2nthrt (problem 3.4.6)

Percentage Accurate: 53.5% → 84.7%

Time: 1.1min

Alternatives: 24

Speedup: 1.8×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.5% 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: 84.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{{n}^{2}}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ t_2 := \frac{t\_1}{n}\\ t_3 := \frac{-0.5 \cdot \frac{{\log x}^{2} - {\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n} - \log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{t\_2}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{t\_2 - t\_1 \cdot \left(\frac{\left(t\_0 - \frac{0.16666666666666666}{{n}^{3}}\right) - \frac{0.3333333333333333}{n}}{{x}^{2}} - \frac{t\_0 + \frac{-0.5}{n}}{x}\right)}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 0.5 (pow n 2.0)))
        (t_1 (pow x (/ 1.0 n)))
        (t_2 (/ t_1 n))
        (t_3
         (/
          (-
           (* -0.5 (/ (- (pow (log x) 2.0) (pow (log1p x) 2.0)) n))
           (log (/ x (+ 1.0 x))))
          n)))
   (if (<= (/ 1.0 n) -1e-160)
     (/ t_2 x)
     (if (<= (/ 1.0 n) 2e-54)
       t_3
       (if (<= (/ 1.0 n) 1e-11)
         (/
          (-
           t_2
           (*
            t_1
            (-
             (/
              (-
               (- t_0 (/ 0.16666666666666666 (pow n 3.0)))
               (/ 0.3333333333333333 n))
              (pow x 2.0))
             (/ (+ t_0 (/ -0.5 n)) x))))
          x)
         (if (<= (/ 1.0 n) 2e-10) t_3 (- (exp (/ (log1p x) n)) t_1)))))))

C

double code(double x, double n) {
	double t_0 = 0.5 / pow(n, 2.0);
	double t_1 = pow(x, (1.0 / n));
	double t_2 = t_1 / n;
	double t_3 = ((-0.5 * ((pow(log(x), 2.0) - pow(log1p(x), 2.0)) / n)) - log((x / (1.0 + x)))) / n;
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = t_2 / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = t_3;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = (t_2 - (t_1 * ((((t_0 - (0.16666666666666666 / pow(n, 3.0))) - (0.3333333333333333 / n)) / pow(x, 2.0)) - ((t_0 + (-0.5 / n)) / x)))) / x;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = t_3;
	} else {
		tmp = exp((log1p(x) / n)) - t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = 0.5 / Math.pow(n, 2.0);
	double t_1 = Math.pow(x, (1.0 / n));
	double t_2 = t_1 / n;
	double t_3 = ((-0.5 * ((Math.pow(Math.log(x), 2.0) - Math.pow(Math.log1p(x), 2.0)) / n)) - Math.log((x / (1.0 + x)))) / n;
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = t_2 / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = t_3;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = (t_2 - (t_1 * ((((t_0 - (0.16666666666666666 / Math.pow(n, 3.0))) - (0.3333333333333333 / n)) / Math.pow(x, 2.0)) - ((t_0 + (-0.5 / n)) / x)))) / x;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = t_3;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 0.5 / math.pow(n, 2.0)
	t_1 = math.pow(x, (1.0 / n))
	t_2 = t_1 / n
	t_3 = ((-0.5 * ((math.pow(math.log(x), 2.0) - math.pow(math.log1p(x), 2.0)) / n)) - math.log((x / (1.0 + x)))) / n
	tmp = 0
	if (1.0 / n) <= -1e-160:
		tmp = t_2 / x
	elif (1.0 / n) <= 2e-54:
		tmp = t_3
	elif (1.0 / n) <= 1e-11:
		tmp = (t_2 - (t_1 * ((((t_0 - (0.16666666666666666 / math.pow(n, 3.0))) - (0.3333333333333333 / n)) / math.pow(x, 2.0)) - ((t_0 + (-0.5 / n)) / x)))) / x
	elif (1.0 / n) <= 2e-10:
		tmp = t_3
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64(0.5 / (n ^ 2.0))
	t_1 = x ^ Float64(1.0 / n)
	t_2 = Float64(t_1 / n)
	t_3 = Float64(Float64(Float64(-0.5 * Float64(Float64((log(x) ^ 2.0) - (log1p(x) ^ 2.0)) / n)) - log(Float64(x / Float64(1.0 + x)))) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-160)
		tmp = Float64(t_2 / x);
	elseif (Float64(1.0 / n) <= 2e-54)
		tmp = t_3;
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(Float64(t_2 - Float64(t_1 * Float64(Float64(Float64(Float64(t_0 - Float64(0.16666666666666666 / (n ^ 3.0))) - Float64(0.3333333333333333 / n)) / (x ^ 2.0)) - Float64(Float64(t_0 + Float64(-0.5 / n)) / x)))) / x);
	elseif (Float64(1.0 / n) <= 2e-10)
		tmp = t_3;
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_1);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(0.5 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / n), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-0.5 * N[(N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-160], N[(t$95$2 / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-54], t$95$3, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[(t$95$2 - N[(t$95$1 * N[(N[(N[(N[(t$95$0 - N[(0.16666666666666666 / N[Power[n, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.3333333333333333 / n), $MachinePrecision]), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$0 + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-10], t$95$3, N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999999e-161

   1. Initial program 74.0%

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

   3. Taylor expanded in x around inf 88.5%

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

      1. associate-/r* 88.5%

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

      2. mul-1-neg 88.5%

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

      3. log-rec 88.5%

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

      4. mul-1-neg 88.5%

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

      5. distribute-neg-frac 88.5%

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

      6. mul-1-neg 88.5%

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

      7. remove-double-neg 88.5%

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

      8. *-rgt-identity 88.5%

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

      9. associate-/l* 88.5%

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

      10. exp-to-pow 88.5%

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

   5. Simplified 88.5%

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

   if -9.9999999999999999e-161 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-54 or 9.99999999999999939e-12 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000007e-10

   1. Initial program 33.7%

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

   3. Taylor expanded in n around -inf 90.4%

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

      1. mul-1-neg 90.4%

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

      2. distribute-neg-frac2 90.4%

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

   5. Simplified 90.4%

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

      1. log1p-undefine 90.4%

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

      2. diff-log 90.5%

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

   7. Applied egg-rr 90.5%

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

      1. +-commutative 90.5%

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

   9. Simplified 90.5%

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

   if 2.0000000000000001e-54 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999939e-12

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf 85.6%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(0.16666666666666666 \cdot \frac{1}{{n}^{3}} + 0.3333333333333333 \cdot \frac{1}{n}\right) - 0.5 \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n} + {x}^{\left(\frac{1}{n}\right)} \cdot \left(\frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{x} + \frac{\frac{0.3333333333333333}{n} + \left(\frac{0.16666666666666666}{{n}^{3}} - \frac{0.5}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   if 2.00000000000000007e-10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 43.1%

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

   3. Taylor expanded in n around 0 43.1%

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

      1. log1p-define 95.5%

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

      2. *-rgt-identity 95.5%

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

      3. associate-/l* 95.5%

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

      4. exp-to-pow 95.5%

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

   5. Simplified 95.5%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{{\log x}^{2} - {\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n} - \log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n} - {x}^{\left(\frac{1}{n}\right)} \cdot \left(\frac{\left(\frac{0.5}{{n}^{2}} - \frac{0.16666666666666666}{{n}^{3}}\right) - \frac{0.3333333333333333}{n}}{{x}^{2}} - \frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{x}\right)}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{{\log x}^{2} - {\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n} - \log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{{n}^{2}}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ t_2 := \frac{t\_1}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{t\_2}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{{\log x}^{2} - {\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n} - \log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \frac{t\_0 + \frac{-0.5}{n}}{x}, t\_2\right) - t\_1 \cdot \left(\frac{\left(t\_0 - \frac{0.16666666666666666}{{n}^{3}}\right) - \frac{0.3333333333333333}{n}}{{x}^{2}} - \frac{\frac{0.041666666666666664}{{n}^{4}} + \left(\left(\frac{0.4583333333333333}{{n}^{2}} + \frac{-0.25}{n}\right) + \frac{-0.25}{{n}^{3}}\right)}{{x}^{3}}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 0.5 (pow n 2.0))) (t_1 (pow x (/ 1.0 n))) (t_2 (/ t_1 n)))
   (if (<= (/ 1.0 n) -1e-160)
     (/ t_2 x)
     (if (<= (/ 1.0 n) 2e-54)
       (/
        (-
         (* -0.5 (/ (- (pow (log x) 2.0) (pow (log1p x) 2.0)) n))
         (log (/ x (+ 1.0 x))))
        n)
       (if (<= (/ 1.0 n) 1e-5)
         (/
          (-
           (fma t_1 (/ (+ t_0 (/ -0.5 n)) x) t_2)
           (*
            t_1
            (-
             (/
              (-
               (- t_0 (/ 0.16666666666666666 (pow n 3.0)))
               (/ 0.3333333333333333 n))
              (pow x 2.0))
             (/
              (+
               (/ 0.041666666666666664 (pow n 4.0))
               (+
                (+ (/ 0.4583333333333333 (pow n 2.0)) (/ -0.25 n))
                (/ -0.25 (pow n 3.0))))
              (pow x 3.0)))))
          x)
         (- (exp (/ (log1p x) n)) t_1))))))

C

double code(double x, double n) {
	double t_0 = 0.5 / pow(n, 2.0);
	double t_1 = pow(x, (1.0 / n));
	double t_2 = t_1 / n;
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = t_2 / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = ((-0.5 * ((pow(log(x), 2.0) - pow(log1p(x), 2.0)) / n)) - log((x / (1.0 + x)))) / n;
	} else if ((1.0 / n) <= 1e-5) {
		tmp = (fma(t_1, ((t_0 + (-0.5 / n)) / x), t_2) - (t_1 * ((((t_0 - (0.16666666666666666 / pow(n, 3.0))) - (0.3333333333333333 / n)) / pow(x, 2.0)) - (((0.041666666666666664 / pow(n, 4.0)) + (((0.4583333333333333 / pow(n, 2.0)) + (-0.25 / n)) + (-0.25 / pow(n, 3.0)))) / pow(x, 3.0))))) / x;
	} else {
		tmp = exp((log1p(x) / n)) - t_1;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(0.5 / (n ^ 2.0))
	t_1 = x ^ Float64(1.0 / n)
	t_2 = Float64(t_1 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-160)
		tmp = Float64(t_2 / x);
	elseif (Float64(1.0 / n) <= 2e-54)
		tmp = Float64(Float64(Float64(-0.5 * Float64(Float64((log(x) ^ 2.0) - (log1p(x) ^ 2.0)) / n)) - log(Float64(x / Float64(1.0 + x)))) / n);
	elseif (Float64(1.0 / n) <= 1e-5)
		tmp = Float64(Float64(fma(t_1, Float64(Float64(t_0 + Float64(-0.5 / n)) / x), t_2) - Float64(t_1 * Float64(Float64(Float64(Float64(t_0 - Float64(0.16666666666666666 / (n ^ 3.0))) - Float64(0.3333333333333333 / n)) / (x ^ 2.0)) - Float64(Float64(Float64(0.041666666666666664 / (n ^ 4.0)) + Float64(Float64(Float64(0.4583333333333333 / (n ^ 2.0)) + Float64(-0.25 / n)) + Float64(-0.25 / (n ^ 3.0)))) / (x ^ 3.0))))) / x);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_1);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(0.5 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-160], N[(t$95$2 / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-54], N[(N[(N[(-0.5 * N[(N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-5], N[(N[(N[(t$95$1 * N[(N[(t$95$0 + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + t$95$2), $MachinePrecision] - N[(t$95$1 * N[(N[(N[(N[(t$95$0 - N[(0.16666666666666666 / N[Power[n, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.3333333333333333 / n), $MachinePrecision]), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.041666666666666664 / N[Power[n, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.4583333333333333 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.25 / n), $MachinePrecision]), $MachinePrecision] + N[(-0.25 / N[Power[n, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999999e-161

   1. Initial program 74.0%

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

   3. Taylor expanded in x around inf 88.5%

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

      1. associate-/r* 88.5%

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

      2. mul-1-neg 88.5%

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

      3. log-rec 88.5%

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

      4. mul-1-neg 88.5%

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

      5. distribute-neg-frac 88.5%

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

      6. mul-1-neg 88.5%

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

      7. remove-double-neg 88.5%

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

      8. *-rgt-identity 88.5%

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

      9. associate-/l* 88.5%

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

      10. exp-to-pow 88.5%

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

   5. Simplified 88.5%

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

   if -9.9999999999999999e-161 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-54

   1. Initial program 33.7%

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

   3. Taylor expanded in n around -inf 90.5%

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

      1. mul-1-neg 90.5%

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

      2. distribute-neg-frac2 90.5%

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

   5. Simplified 90.5%

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

      1. log1p-undefine 90.5%

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

      2. diff-log 90.5%

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

   7. Applied egg-rr 90.5%

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

      1. +-commutative 90.5%

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

   9. Simplified 90.5%

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

   if 2.0000000000000001e-54 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000008e-5

   1. Initial program 15.6%

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

   3. Taylor expanded in x around inf 77.3%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(0.041666666666666664 \cdot \frac{1}{{n}^{4}} + 0.4583333333333333 \cdot \frac{1}{{n}^{2}}\right) - \left(0.25 \cdot \frac{1}{n} + 0.25 \cdot \frac{1}{{n}^{3}}\right)\right)}{{x}^{3}} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(0.16666666666666666 \cdot \frac{1}{{n}^{3}} + 0.3333333333333333 \cdot \frac{1}{n}\right) - 0.5 \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)\right)}{x}} \]

   5. Simplified 77.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right) + {x}^{\left(\frac{1}{n}\right)} \cdot \left(\frac{\frac{0.3333333333333333}{n} + \left(\frac{0.16666666666666666}{{n}^{3}} - \frac{0.5}{{n}^{2}}\right)}{{x}^{2}} + \frac{\frac{0.041666666666666664}{{n}^{4}} + \left(\left(\frac{0.4583333333333333}{{n}^{2}} + \frac{-0.25}{n}\right) + \frac{-0.25}{{n}^{3}}\right)}{{x}^{3}}\right)}{x}} \]

   if 1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 43.0%

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

   3. Taylor expanded in n around 0 43.0%

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

      1. log1p-define 97.7%

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

      2. *-rgt-identity 97.7%

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

      3. associate-/l* 97.7%

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

      4. exp-to-pow 97.7%

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

   5. Simplified 97.7%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{{\log x}^{2} - {\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n} - \log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right) - {x}^{\left(\frac{1}{n}\right)} \cdot \left(\frac{\left(\frac{0.5}{{n}^{2}} - \frac{0.16666666666666666}{{n}^{3}}\right) - \frac{0.3333333333333333}{n}}{{x}^{2}} - \frac{\frac{0.041666666666666664}{{n}^{4}} + \left(\left(\frac{0.4583333333333333}{{n}^{2}} + \frac{-0.25}{n}\right) + \frac{-0.25}{{n}^{3}}\right)}{{x}^{3}}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{0.5}{{n}^{2}}\\ \mathbf{if}\;n \leq -5.6 \cdot 10^{+159}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;n \leq -13000:\\ \;\;\;\;\frac{\frac{t\_0}{n} - t\_0 \cdot \left(\frac{\left(t\_1 - \frac{0.16666666666666666}{{n}^{3}}\right) - \frac{0.3333333333333333}{n}}{{x}^{2}} - \frac{t\_1 + \frac{-0.5}{n}}{x}\right)}{x}\\ \mathbf{elif}\;n \leq 8500000:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right) + -0.5 \cdot \log \left(e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ 0.5 (pow n 2.0))))
   (if (<= n -5.6e+159)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= n -13000.0)
       (/
        (-
         (/ t_0 n)
         (*
          t_0
          (-
           (/
            (-
             (- t_1 (/ 0.16666666666666666 (pow n 3.0)))
             (/ 0.3333333333333333 n))
            (pow x 2.0))
           (/ (+ t_1 (/ -0.5 n)) x))))
        x)
       (if (<= n 8500000.0)
         (- (exp (/ (log1p x) n)) t_0)
         (/
          (+
           (log (/ x (+ 1.0 x)))
           (*
            -0.5
            (log (exp (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n)))))
          (- n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = 0.5 / pow(n, 2.0);
	double tmp;
	if (n <= -5.6e+159) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if (n <= -13000.0) {
		tmp = ((t_0 / n) - (t_0 * ((((t_1 - (0.16666666666666666 / pow(n, 3.0))) - (0.3333333333333333 / n)) / pow(x, 2.0)) - ((t_1 + (-0.5 / n)) / x)))) / x;
	} else if (n <= 8500000.0) {
		tmp = exp((log1p(x) / n)) - t_0;
	} else {
		tmp = (log((x / (1.0 + x))) + (-0.5 * log(exp(((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n))))) / -n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = 0.5 / Math.pow(n, 2.0);
	double tmp;
	if (n <= -5.6e+159) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if (n <= -13000.0) {
		tmp = ((t_0 / n) - (t_0 * ((((t_1 - (0.16666666666666666 / Math.pow(n, 3.0))) - (0.3333333333333333 / n)) / Math.pow(x, 2.0)) - ((t_1 + (-0.5 / n)) / x)))) / x;
	} else if (n <= 8500000.0) {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	} else {
		tmp = (Math.log((x / (1.0 + x))) + (-0.5 * Math.log(Math.exp(((Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0)) / n))))) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = 0.5 / math.pow(n, 2.0)
	tmp = 0
	if n <= -5.6e+159:
		tmp = math.log(((1.0 + x) / x)) / n
	elif n <= -13000.0:
		tmp = ((t_0 / n) - (t_0 * ((((t_1 - (0.16666666666666666 / math.pow(n, 3.0))) - (0.3333333333333333 / n)) / math.pow(x, 2.0)) - ((t_1 + (-0.5 / n)) / x)))) / x
	elif n <= 8500000.0:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	else:
		tmp = (math.log((x / (1.0 + x))) + (-0.5 * math.log(math.exp(((math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0)) / n))))) / -n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(0.5 / (n ^ 2.0))
	tmp = 0.0
	if (n <= -5.6e+159)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (n <= -13000.0)
		tmp = Float64(Float64(Float64(t_0 / n) - Float64(t_0 * Float64(Float64(Float64(Float64(t_1 - Float64(0.16666666666666666 / (n ^ 3.0))) - Float64(0.3333333333333333 / n)) / (x ^ 2.0)) - Float64(Float64(t_1 + Float64(-0.5 / n)) / x)))) / x);
	elseif (n <= 8500000.0)
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	else
		tmp = Float64(Float64(log(Float64(x / Float64(1.0 + x))) + Float64(-0.5 * log(exp(Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / n))))) / Float64(-n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5.6e+159], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -13000.0], N[(N[(N[(t$95$0 / n), $MachinePrecision] - N[(t$95$0 * N[(N[(N[(N[(t$95$1 - N[(0.16666666666666666 / N[Power[n, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.3333333333333333 / n), $MachinePrecision]), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[n, 8500000.0], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[Log[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]), $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.6000000000000002e159

   1. Initial program 28.0%

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

   3. Taylor expanded in n around inf 93.9%

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

      1. log1p-define 93.9%

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

   5. Simplified 93.9%

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

      1. log1p-undefine 93.9%

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

      2. +-commutative 93.9%

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

      3. diff-log 93.9%

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

   7. Applied egg-rr 93.9%

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

   if -5.6000000000000002e159 < n < -13000

   1. Initial program 16.0%

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

   3. Taylor expanded in x around inf 65.7%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(0.16666666666666666 \cdot \frac{1}{{n}^{3}} + 0.3333333333333333 \cdot \frac{1}{n}\right) - 0.5 \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 65.7%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n} + {x}^{\left(\frac{1}{n}\right)} \cdot \left(\frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{x} + \frac{\frac{0.3333333333333333}{n} + \left(\frac{0.16666666666666666}{{n}^{3}} - \frac{0.5}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   if -13000 < n < 8.5e6

   1. Initial program 78.2%

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

   3. Taylor expanded in n around 0 78.2%

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

      1. log1p-define 98.3%

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

      2. *-rgt-identity 98.3%

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

      3. associate-/l* 98.3%

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

      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%

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

   if 8.5e6 < n

   1. Initial program 33.4%

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

   3. Taylor expanded in n around -inf 82.2%

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

      1. mul-1-neg 82.2%

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

      2. distribute-neg-frac2 82.2%

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

   5. Simplified 82.2%

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

      1. log1p-undefine 82.2%

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

      2. diff-log 82.4%

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

   7. Applied egg-rr 82.4%

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

      1. +-commutative 82.4%

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

   9. Simplified 82.4%

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

       1. add-log-exp 82.5%

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

   11. Applied egg-rr 82.5%

       \[\leadsto \frac{-0.5 \cdot \color{blue}{\log \left(e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}\right)} + \log \left(\frac{x}{x + 1}\right)}{-n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.6 \cdot 10^{+159}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;n \leq -13000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n} - {x}^{\left(\frac{1}{n}\right)} \cdot \left(\frac{\left(\frac{0.5}{{n}^{2}} - \frac{0.16666666666666666}{{n}^{3}}\right) - \frac{0.3333333333333333}{n}}{{x}^{2}} - \frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{x}\right)}{x}\\ \mathbf{elif}\;n \leq 8500000:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right) + -0.5 \cdot \log \left(e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{-0.5 \cdot \frac{{\log x}^{2} - {\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n} - \log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n} - \frac{\frac{0.5}{x} + -1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1
         (/
          (-
           (* -0.5 (/ (- (pow (log x) 2.0) (pow (log1p x) 2.0)) n))
           (log (/ x (+ 1.0 x))))
          n)))
   (if (<= (/ 1.0 n) -1e-160)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-54)
       t_1
       (if (<= (/ 1.0 n) 1e-11)
         (/
          (- (* -0.5 (/ (/ (* (log x) -2.0) x) n)) (/ (+ (/ 0.5 x) -1.0) x))
          n)
         (if (<= (/ 1.0 n) 2e-10) t_1 (- (exp (/ (log1p x) n)) t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = ((-0.5 * ((pow(log(x), 2.0) - pow(log1p(x), 2.0)) / n)) - log((x / (1.0 + x)))) / n;
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = ((-0.5 * (((log(x) * -2.0) / x) / n)) - (((0.5 / x) + -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = t_1;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = ((-0.5 * ((Math.pow(Math.log(x), 2.0) - Math.pow(Math.log1p(x), 2.0)) / n)) - Math.log((x / (1.0 + x)))) / n;
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = ((-0.5 * (((Math.log(x) * -2.0) / x) / n)) - (((0.5 / x) + -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = t_1;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = ((-0.5 * ((math.pow(math.log(x), 2.0) - math.pow(math.log1p(x), 2.0)) / n)) - math.log((x / (1.0 + x)))) / n
	tmp = 0
	if (1.0 / n) <= -1e-160:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-54:
		tmp = t_1
	elif (1.0 / n) <= 1e-11:
		tmp = ((-0.5 * (((math.log(x) * -2.0) / x) / n)) - (((0.5 / x) + -1.0) / x)) / n
	elif (1.0 / n) <= 2e-10:
		tmp = t_1
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(Float64(-0.5 * Float64(Float64((log(x) ^ 2.0) - (log1p(x) ^ 2.0)) / n)) - log(Float64(x / Float64(1.0 + x)))) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-160)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-54)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(Float64(Float64(-0.5 * Float64(Float64(Float64(log(x) * -2.0) / x) / n)) - Float64(Float64(Float64(0.5 / x) + -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e-10)
		tmp = t_1;
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-0.5 * N[(N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-160], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-54], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[(N[(-0.5 * N[(N[(N[(N[Log[x], $MachinePrecision] * -2.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.5 / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-10], t$95$1, N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999999e-161

   1. Initial program 74.0%

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

   3. Taylor expanded in x around inf 88.5%

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

      1. associate-/r* 88.5%

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

      2. mul-1-neg 88.5%

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

      3. log-rec 88.5%

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

      4. mul-1-neg 88.5%

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

      5. distribute-neg-frac 88.5%

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

      6. mul-1-neg 88.5%

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

      7. remove-double-neg 88.5%

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

      8. *-rgt-identity 88.5%

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

      9. associate-/l* 88.5%

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

      10. exp-to-pow 88.5%

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

   5. Simplified 88.5%

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

   if -9.9999999999999999e-161 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-54 or 9.99999999999999939e-12 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000007e-10

   1. Initial program 33.7%

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

   3. Taylor expanded in n around -inf 90.4%

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

      1. mul-1-neg 90.4%

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

      2. distribute-neg-frac2 90.4%

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

   5. Simplified 90.4%

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

      1. log1p-undefine 90.4%

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

      2. diff-log 90.5%

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

   7. Applied egg-rr 90.5%

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

      1. +-commutative 90.5%

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

   9. Simplified 90.5%

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

   if 2.0000000000000001e-54 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999939e-12

   1. Initial program 4.8%

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

   3. Taylor expanded in n around -inf 26.2%

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

      1. mul-1-neg 26.2%

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

      2. distribute-neg-frac2 26.2%

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

   5. Simplified 26.2%

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

   6. Taylor expanded in x around inf 81.0%

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

      1. sub-neg 81.0%

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

      2. associate-*r/ 81.0%

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

      3. metadata-eval 81.0%

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

      4. metadata-eval 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in x around inf 84.7%

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

       1. associate-*r/ 84.7%

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

       2. *-commutative 84.7%

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

       3. log-rec 84.7%

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

   11. Simplified 84.7%

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

   if 2.00000000000000007e-10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 43.1%

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

   3. Taylor expanded in n around 0 43.1%

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

      1. log1p-define 95.5%

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

      2. *-rgt-identity 95.5%

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

      3. associate-/l* 95.5%

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

      4. exp-to-pow 95.5%

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

   5. Simplified 95.5%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{{\log x}^{2} - {\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n} - \log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n} - \frac{\frac{0.5}{x} + -1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{{\log x}^{2} - {\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n} - \log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{1}{x}\right)\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{t\_1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n} - \frac{\frac{0.5}{x} + -1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\left(\frac{-1}{x} - \frac{t\_0}{x}\right) - -2 \cdot t\_0}{x}}{n} - \log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (log (/ 1.0 x))) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-160)
     (/ (/ t_1 n) x)
     (if (<= (/ 1.0 n) 2e-54)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 1e-11)
         (/
          (- (* -0.5 (/ (/ (* (log x) -2.0) x) n)) (/ (+ (/ 0.5 x) -1.0) x))
          n)
         (if (<= (/ 1.0 n) 2e-10)
           (/
            (-
             (* -0.5 (/ (/ (- (- (/ -1.0 x) (/ t_0 x)) (* -2.0 t_0)) x) n))
             (log (/ x (+ 1.0 x))))
            n)
           (- (exp (/ (log1p x) n)) t_1)))))))

C

double code(double x, double n) {
	double t_0 = log((1.0 / x));
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = ((-0.5 * (((log(x) * -2.0) / x) / n)) - (((0.5 / x) + -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = ((-0.5 * (((((-1.0 / x) - (t_0 / x)) - (-2.0 * t_0)) / x) / n)) - log((x / (1.0 + x)))) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log((1.0 / x));
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = ((-0.5 * (((Math.log(x) * -2.0) / x) / n)) - (((0.5 / x) + -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = ((-0.5 * (((((-1.0 / x) - (t_0 / x)) - (-2.0 * t_0)) / x) / n)) - Math.log((x / (1.0 + x)))) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log((1.0 / x))
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-160:
		tmp = (t_1 / n) / x
	elif (1.0 / n) <= 2e-54:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 1e-11:
		tmp = ((-0.5 * (((math.log(x) * -2.0) / x) / n)) - (((0.5 / x) + -1.0) / x)) / n
	elif (1.0 / n) <= 2e-10:
		tmp = ((-0.5 * (((((-1.0 / x) - (t_0 / x)) - (-2.0 * t_0)) / x) / n)) - math.log((x / (1.0 + x)))) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_1
	return tmp

Julia

function code(x, n)
	t_0 = log(Float64(1.0 / x))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-160)
		tmp = Float64(Float64(t_1 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-54)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(Float64(Float64(-0.5 * Float64(Float64(Float64(log(x) * -2.0) / x) / n)) - Float64(Float64(Float64(0.5 / x) + -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e-10)
		tmp = Float64(Float64(Float64(-0.5 * Float64(Float64(Float64(Float64(Float64(-1.0 / x) - Float64(t_0 / x)) - Float64(-2.0 * t_0)) / x) / n)) - log(Float64(x / Float64(1.0 + x)))) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_1);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-160], N[(N[(t$95$1 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-54], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[(N[(-0.5 * N[(N[(N[(N[Log[x], $MachinePrecision] * -2.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.5 / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-10], N[(N[(N[(-0.5 * N[(N[(N[(N[(N[(-1.0 / x), $MachinePrecision] - N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision] - N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999999e-161

   1. Initial program 74.0%

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

   3. Taylor expanded in x around inf 88.5%

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

      1. associate-/r* 88.5%

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

      2. mul-1-neg 88.5%

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

      3. log-rec 88.5%

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

      4. mul-1-neg 88.5%

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

      5. distribute-neg-frac 88.5%

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

      6. mul-1-neg 88.5%

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

      7. remove-double-neg 88.5%

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

      8. *-rgt-identity 88.5%

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

      9. associate-/l* 88.5%

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

      10. exp-to-pow 88.5%

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

   5. Simplified 88.5%

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

   if -9.9999999999999999e-161 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-54

   1. Initial program 33.7%

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

   3. Taylor expanded in n around inf 90.5%

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

      1. log1p-define 90.5%

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

   5. Simplified 90.5%

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

      1. log1p-undefine 90.5%

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

      2. +-commutative 90.5%

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

      3. diff-log 90.5%

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

   7. Applied egg-rr 90.5%

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

   if 2.0000000000000001e-54 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999939e-12

   1. Initial program 4.8%

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

   3. Taylor expanded in n around -inf 26.2%

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

      1. mul-1-neg 26.2%

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

      2. distribute-neg-frac2 26.2%

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

   5. Simplified 26.2%

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

   6. Taylor expanded in x around inf 81.0%

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

      1. sub-neg 81.0%

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

      2. associate-*r/ 81.0%

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

      3. metadata-eval 81.0%

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

      4. metadata-eval 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in x around inf 84.7%

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

       1. associate-*r/ 84.7%

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

       2. *-commutative 84.7%

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

       3. log-rec 84.7%

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

   11. Simplified 84.7%

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

   if 9.99999999999999939e-12 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000007e-10

   1. Initial program 33.8%

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

   3. Taylor expanded in n around -inf 82.2%

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

      1. mul-1-neg 82.2%

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

      2. distribute-neg-frac2 82.2%

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

   5. Simplified 82.9%

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

      1. log1p-undefine 82.9%

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

      2. diff-log 88.7%

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

   7. Applied egg-rr 88.7%

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

      1. +-commutative 88.7%

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

   9. Simplified 88.7%

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

   10. Taylor expanded in x around inf 88.6%

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

   if 2.00000000000000007e-10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 43.1%

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

   3. Taylor expanded in n around 0 43.1%

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

      1. log1p-define 95.5%

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

      2. *-rgt-identity 95.5%

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

      3. associate-/l* 95.5%

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

      4. exp-to-pow 95.5%

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

   5. Simplified 95.5%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n} - \frac{\frac{0.5}{x} + -1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\left(\frac{-1}{x} - \frac{\log \left(\frac{1}{x}\right)}{x}\right) - -2 \cdot \log \left(\frac{1}{x}\right)}{x}}{n} - \log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{t\_1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{t\_0 - \frac{\frac{0.5}{x} + -1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_0 + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ (/ (* (log x) -2.0) x) n))) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-160)
     (/ (/ t_1 n) x)
     (if (<= (/ 1.0 n) 2e-54)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 1e-11)
         (/ (- t_0 (/ (+ (/ 0.5 x) -1.0) x)) n)
         (if (<= (/ 1.0 n) 2e-10)
           (/ (+ t_0 (- (log1p x) (log x))) n)
           (- (exp (/ (log1p x) n)) t_1)))))))

C

double code(double x, double n) {
	double t_0 = -0.5 * (((log(x) * -2.0) / x) / n);
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = (t_0 - (((0.5 / x) + -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = (t_0 + (log1p(x) - log(x))) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = -0.5 * (((Math.log(x) * -2.0) / x) / n);
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = (t_0 - (((0.5 / x) + -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = (t_0 + (Math.log1p(x) - Math.log(x))) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -0.5 * (((math.log(x) * -2.0) / x) / n)
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-160:
		tmp = (t_1 / n) / x
	elif (1.0 / n) <= 2e-54:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 1e-11:
		tmp = (t_0 - (((0.5 / x) + -1.0) / x)) / n
	elif (1.0 / n) <= 2e-10:
		tmp = (t_0 + (math.log1p(x) - math.log(x))) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64(-0.5 * Float64(Float64(Float64(log(x) * -2.0) / x) / n))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-160)
		tmp = Float64(Float64(t_1 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-54)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(0.5 / x) + -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e-10)
		tmp = Float64(Float64(t_0 + Float64(log1p(x) - log(x))) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_1);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(-0.5 * N[(N[(N[(N[Log[x], $MachinePrecision] * -2.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-160], N[(N[(t$95$1 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-54], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[(t$95$0 - N[(N[(N[(0.5 / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-10], N[(N[(t$95$0 + N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999999e-161

   1. Initial program 74.0%

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

   3. Taylor expanded in x around inf 88.5%

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

      1. associate-/r* 88.5%

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

      2. mul-1-neg 88.5%

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

      3. log-rec 88.5%

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

      4. mul-1-neg 88.5%

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

      5. distribute-neg-frac 88.5%

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

      6. mul-1-neg 88.5%

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

      7. remove-double-neg 88.5%

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

      8. *-rgt-identity 88.5%

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

      9. associate-/l* 88.5%

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

      10. exp-to-pow 88.5%

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

   5. Simplified 88.5%

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

   if -9.9999999999999999e-161 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-54

   1. Initial program 33.7%

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

   3. Taylor expanded in n around inf 90.5%

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

      1. log1p-define 90.5%

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

   5. Simplified 90.5%

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

      1. log1p-undefine 90.5%

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

      2. +-commutative 90.5%

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

      3. diff-log 90.5%

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

   7. Applied egg-rr 90.5%

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

   if 2.0000000000000001e-54 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999939e-12

   1. Initial program 4.8%

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

   3. Taylor expanded in n around -inf 26.2%

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

      1. mul-1-neg 26.2%

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

      2. distribute-neg-frac2 26.2%

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

   5. Simplified 26.2%

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

   6. Taylor expanded in x around inf 81.0%

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

      1. sub-neg 81.0%

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

      2. associate-*r/ 81.0%

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

      3. metadata-eval 81.0%

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

      4. metadata-eval 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in x around inf 84.7%

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

       1. associate-*r/ 84.7%

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

       2. *-commutative 84.7%

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

       3. log-rec 84.7%

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

   11. Simplified 84.7%

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

   if 9.99999999999999939e-12 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000007e-10

   1. Initial program 33.8%

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

   3. Taylor expanded in n around -inf 82.2%

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

      1. mul-1-neg 82.2%

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

      2. distribute-neg-frac2 82.2%

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

   5. Simplified 82.9%

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

   6. Taylor expanded in x around inf 82.0%

      \[\leadsto \frac{-0.5 \cdot \frac{\color{blue}{-2 \cdot \frac{\log \left(\frac{1}{x}\right)}{x}}}{n} + \left(\log x - \mathsf{log1p}\left(x\right)\right)}{-n} \]
   7. Step-by-step derivation

      1. associate-*r/ 47.3%

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

      2. *-commutative 47.3%

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

      3. log-rec 47.3%

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

   8. Simplified 82.0%

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

   if 2.00000000000000007e-10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 43.1%

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

   3. Taylor expanded in n around 0 43.1%

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

      1. log1p-define 95.5%

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

      2. *-rgt-identity 95.5%

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

      3. associate-/l* 95.5%

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

      4. exp-to-pow 95.5%

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

   5. Simplified 95.5%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n} - \frac{\frac{0.5}{x} + -1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{t\_1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{t\_0 - \frac{\frac{0.5}{x} + -1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_0 - \log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ (/ (* (log x) -2.0) x) n))) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-160)
     (/ (/ t_1 n) x)
     (if (<= (/ 1.0 n) 2e-54)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 1e-11)
         (/ (- t_0 (/ (+ (/ 0.5 x) -1.0) x)) n)
         (if (<= (/ 1.0 n) 2e-10)
           (/ (- t_0 (log (/ x (+ 1.0 x)))) n)
           (- (exp (/ (log1p x) n)) t_1)))))))

C

double code(double x, double n) {
	double t_0 = -0.5 * (((log(x) * -2.0) / x) / n);
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = (t_0 - (((0.5 / x) + -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = (t_0 - log((x / (1.0 + x)))) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = -0.5 * (((Math.log(x) * -2.0) / x) / n);
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = (t_0 - (((0.5 / x) + -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = (t_0 - Math.log((x / (1.0 + x)))) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -0.5 * (((math.log(x) * -2.0) / x) / n)
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-160:
		tmp = (t_1 / n) / x
	elif (1.0 / n) <= 2e-54:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 1e-11:
		tmp = (t_0 - (((0.5 / x) + -1.0) / x)) / n
	elif (1.0 / n) <= 2e-10:
		tmp = (t_0 - math.log((x / (1.0 + x)))) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64(-0.5 * Float64(Float64(Float64(log(x) * -2.0) / x) / n))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-160)
		tmp = Float64(Float64(t_1 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-54)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(0.5 / x) + -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e-10)
		tmp = Float64(Float64(t_0 - log(Float64(x / Float64(1.0 + x)))) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_1);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(-0.5 * N[(N[(N[(N[Log[x], $MachinePrecision] * -2.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-160], N[(N[(t$95$1 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-54], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[(t$95$0 - N[(N[(N[(0.5 / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-10], N[(N[(t$95$0 - N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999999e-161

   1. Initial program 74.0%

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

   3. Taylor expanded in x around inf 88.5%

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

      1. associate-/r* 88.5%

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

      2. mul-1-neg 88.5%

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

      3. log-rec 88.5%

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

      4. mul-1-neg 88.5%

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

      5. distribute-neg-frac 88.5%

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

      6. mul-1-neg 88.5%

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

      7. remove-double-neg 88.5%

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

      8. *-rgt-identity 88.5%

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

      9. associate-/l* 88.5%

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

      10. exp-to-pow 88.5%

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

   5. Simplified 88.5%

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

   if -9.9999999999999999e-161 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-54

   1. Initial program 33.7%

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

   3. Taylor expanded in n around inf 90.5%

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

      1. log1p-define 90.5%

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

   5. Simplified 90.5%

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

      1. log1p-undefine 90.5%

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

      2. +-commutative 90.5%

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

      3. diff-log 90.5%

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

   7. Applied egg-rr 90.5%

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

   if 2.0000000000000001e-54 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999939e-12

   1. Initial program 4.8%

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

   3. Taylor expanded in n around -inf 26.2%

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

      1. mul-1-neg 26.2%

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

      2. distribute-neg-frac2 26.2%

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

   5. Simplified 26.2%

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

   6. Taylor expanded in x around inf 81.0%

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

      1. sub-neg 81.0%

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

      2. associate-*r/ 81.0%

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

      3. metadata-eval 81.0%

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

      4. metadata-eval 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in x around inf 84.7%

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

       1. associate-*r/ 84.7%

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

       2. *-commutative 84.7%

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

       3. log-rec 84.7%

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

   11. Simplified 84.7%

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

   if 9.99999999999999939e-12 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000007e-10

   1. Initial program 33.8%

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

   3. Taylor expanded in n around -inf 82.2%

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

      1. mul-1-neg 82.2%

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

      2. distribute-neg-frac2 82.2%

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

   5. Simplified 82.9%

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

      1. log1p-undefine 82.9%

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

      2. diff-log 88.7%

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

   7. Applied egg-rr 88.7%

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

      1. +-commutative 88.7%

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

   9. Simplified 88.7%

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

   10. Taylor expanded in x around inf 80.9%

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

       1. associate-*r/ 47.3%

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

       2. *-commutative 47.3%

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

       3. log-rec 47.3%

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

   12. Simplified 80.9%

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

   if 2.00000000000000007e-10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 43.1%

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

   3. Taylor expanded in n around 0 43.1%

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

      1. log1p-define 95.5%

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

      2. *-rgt-identity 95.5%

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

      3. associate-/l* 95.5%

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

      4. exp-to-pow 95.5%

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

   5. Simplified 95.5%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n} - \frac{\frac{0.5}{x} + -1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n} - \log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{t\_1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{t\_0 - \frac{\frac{0.5}{x} + -1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_0 - \log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x \cdot \left(\frac{-1}{n} - x \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right)\right) - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ (/ (* (log x) -2.0) x) n))) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-160)
     (/ (/ t_1 n) x)
     (if (<= (/ 1.0 n) 2e-54)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 1e-11)
         (/ (- t_0 (/ (+ (/ 0.5 x) -1.0) x)) n)
         (if (<= (/ 1.0 n) 2e-10)
           (/ (- t_0 (log (/ x (+ 1.0 x)))) n)
           (-
            (-
             1.0
             (*
              x
              (-
               (/ -1.0 n)
               (* x (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n)))))))
            t_1)))))))

C

double code(double x, double n) {
	double t_0 = -0.5 * (((log(x) * -2.0) / x) / n);
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = (t_0 - (((0.5 / x) + -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = (t_0 - log((x / (1.0 + x)))) / n;
	} else {
		tmp = (1.0 - (x * ((-1.0 / n) - (x * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n))))))) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.5d0) * (((log(x) * (-2.0d0)) / x) / n)
    t_1 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1d-160)) then
        tmp = (t_1 / n) / x
    else if ((1.0d0 / n) <= 2d-54) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 1d-11) then
        tmp = (t_0 - (((0.5d0 / x) + (-1.0d0)) / x)) / n
    else if ((1.0d0 / n) <= 2d-10) then
        tmp = (t_0 - log((x / (1.0d0 + x)))) / n
    else
        tmp = (1.0d0 - (x * (((-1.0d0) / n) - (x * ((0.5d0 * (1.0d0 / (n ** 2.0d0))) + (0.5d0 * ((-1.0d0) / n))))))) - t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -0.5 * (((Math.log(x) * -2.0) / x) / n);
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = (t_0 - (((0.5 / x) + -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = (t_0 - Math.log((x / (1.0 + x)))) / n;
	} else {
		tmp = (1.0 - (x * ((-1.0 / n) - (x * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))))) - t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -0.5 * (((math.log(x) * -2.0) / x) / n)
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-160:
		tmp = (t_1 / n) / x
	elif (1.0 / n) <= 2e-54:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 1e-11:
		tmp = (t_0 - (((0.5 / x) + -1.0) / x)) / n
	elif (1.0 / n) <= 2e-10:
		tmp = (t_0 - math.log((x / (1.0 + x)))) / n
	else:
		tmp = (1.0 - (x * ((-1.0 / n) - (x * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))))) - t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64(-0.5 * Float64(Float64(Float64(log(x) * -2.0) / x) / n))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-160)
		tmp = Float64(Float64(t_1 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-54)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(0.5 / x) + -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e-10)
		tmp = Float64(Float64(t_0 - log(Float64(x / Float64(1.0 + x)))) / n);
	else
		tmp = Float64(Float64(1.0 - Float64(x * Float64(Float64(-1.0 / n) - Float64(x * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))))) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -0.5 * (((log(x) * -2.0) / x) / n);
	t_1 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-160)
		tmp = (t_1 / n) / x;
	elseif ((1.0 / n) <= 2e-54)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 1e-11)
		tmp = (t_0 - (((0.5 / x) + -1.0) / x)) / n;
	elseif ((1.0 / n) <= 2e-10)
		tmp = (t_0 - log((x / (1.0 + x)))) / n;
	else
		tmp = (1.0 - (x * ((-1.0 / n) - (x * ((0.5 * (1.0 / (n ^ 2.0))) + (0.5 * (-1.0 / n))))))) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(-0.5 * N[(N[(N[(N[Log[x], $MachinePrecision] * -2.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-160], N[(N[(t$95$1 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-54], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[(t$95$0 - N[(N[(N[(0.5 / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-10], N[(N[(t$95$0 - N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 - N[(x * N[(N[(-1.0 / n), $MachinePrecision] - N[(x * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999999e-161

   1. Initial program 74.0%

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

   3. Taylor expanded in x around inf 88.5%

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

      1. associate-/r* 88.5%

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

      2. mul-1-neg 88.5%

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

      3. log-rec 88.5%

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

      4. mul-1-neg 88.5%

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

      5. distribute-neg-frac 88.5%

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

      6. mul-1-neg 88.5%

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

      7. remove-double-neg 88.5%

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

      8. *-rgt-identity 88.5%

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

      9. associate-/l* 88.5%

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

      10. exp-to-pow 88.5%

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

   5. Simplified 88.5%

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

   if -9.9999999999999999e-161 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-54

   1. Initial program 33.7%

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

   3. Taylor expanded in n around inf 90.5%

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

      1. log1p-define 90.5%

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

   5. Simplified 90.5%

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

      1. log1p-undefine 90.5%

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

      2. +-commutative 90.5%

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

      3. diff-log 90.5%

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

   7. Applied egg-rr 90.5%

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

   if 2.0000000000000001e-54 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999939e-12

   1. Initial program 4.8%

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

   3. Taylor expanded in n around -inf 26.2%

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

      1. mul-1-neg 26.2%

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

      2. distribute-neg-frac2 26.2%

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

   5. Simplified 26.2%

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

   6. Taylor expanded in x around inf 81.0%

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

      1. sub-neg 81.0%

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

      2. associate-*r/ 81.0%

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

      3. metadata-eval 81.0%

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

      4. metadata-eval 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in x around inf 84.7%

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

       1. associate-*r/ 84.7%

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

       2. *-commutative 84.7%

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

       3. log-rec 84.7%

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

   11. Simplified 84.7%

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

   if 9.99999999999999939e-12 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000007e-10

   1. Initial program 33.8%

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

   3. Taylor expanded in n around -inf 82.2%

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

      1. mul-1-neg 82.2%

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

      2. distribute-neg-frac2 82.2%

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

   5. Simplified 82.9%

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

      1. log1p-undefine 82.9%

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

      2. diff-log 88.7%

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

   7. Applied egg-rr 88.7%

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

      1. +-commutative 88.7%

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

   9. Simplified 88.7%

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

   10. Taylor expanded in x around inf 80.9%

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

       1. associate-*r/ 47.3%

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

       2. *-commutative 47.3%

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

       3. log-rec 47.3%

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

   12. Simplified 80.9%

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

   if 2.00000000000000007e-10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 43.1%

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

   3. Taylor expanded in x around 0 70.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n} - \frac{\frac{0.5}{x} + -1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n} - \log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x \cdot \left(\frac{-1}{n} - x \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n} - \frac{\frac{0.5}{x} + -1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+181}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{x}}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -1e-160)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-54)
       t_1
       (if (<= (/ 1.0 n) 1e-11)
         (/
          (- (* -0.5 (/ (/ (* (log x) -2.0) x) n)) (/ (+ (/ 0.5 x) -1.0) x))
          n)
         (if (<= (/ 1.0 n) 2e-10)
           t_1
           (if (<= (/ 1.0 n) 5e+181)
             (- (pow (+ 1.0 x) (/ 1.0 n)) t_0)
             (log1p (expm1 (/ (/ 1.0 x) n))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = ((-0.5 * (((log(x) * -2.0) / x) / n)) - (((0.5 / x) + -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+181) {
		tmp = pow((1.0 + x), (1.0 / n)) - t_0;
	} else {
		tmp = log1p(expm1(((1.0 / x) / n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = ((-0.5 * (((Math.log(x) * -2.0) / x) / n)) - (((0.5 / x) + -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+181) {
		tmp = Math.pow((1.0 + x), (1.0 / n)) - t_0;
	} else {
		tmp = Math.log1p(Math.expm1(((1.0 / x) / n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if (1.0 / n) <= -1e-160:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-54:
		tmp = t_1
	elif (1.0 / n) <= 1e-11:
		tmp = ((-0.5 * (((math.log(x) * -2.0) / x) / n)) - (((0.5 / x) + -1.0) / x)) / n
	elif (1.0 / n) <= 2e-10:
		tmp = t_1
	elif (1.0 / n) <= 5e+181:
		tmp = math.pow((1.0 + x), (1.0 / n)) - t_0
	else:
		tmp = math.log1p(math.expm1(((1.0 / x) / n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-160)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-54)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(Float64(Float64(-0.5 * Float64(Float64(Float64(log(x) * -2.0) / x) / n)) - Float64(Float64(Float64(0.5 / x) + -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e-10)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e+181)
		tmp = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - t_0);
	else
		tmp = log1p(expm1(Float64(Float64(1.0 / x) / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-160], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-54], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[(N[(-0.5 * N[(N[(N[(N[Log[x], $MachinePrecision] * -2.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.5 / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-10], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+181], N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], N[Log[1 + N[(Exp[N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999999e-161

   1. Initial program 74.0%

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

   3. Taylor expanded in x around inf 88.5%

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

      1. associate-/r* 88.5%

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

      2. mul-1-neg 88.5%

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

      3. log-rec 88.5%

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

      4. mul-1-neg 88.5%

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

      5. distribute-neg-frac 88.5%

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

      6. mul-1-neg 88.5%

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

      7. remove-double-neg 88.5%

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

      8. *-rgt-identity 88.5%

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

      9. associate-/l* 88.5%

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

      10. exp-to-pow 88.5%

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

   5. Simplified 88.5%

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

   if -9.9999999999999999e-161 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-54 or 9.99999999999999939e-12 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000007e-10

   1. Initial program 33.7%

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

   3. Taylor expanded in n around inf 90.2%

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

      1. log1p-define 90.2%

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

   5. Simplified 90.2%

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

      1. log1p-undefine 90.2%

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

      2. +-commutative 90.2%

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

      3. diff-log 90.2%

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

   7. Applied egg-rr 90.2%

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

   if 2.0000000000000001e-54 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999939e-12

   1. Initial program 4.8%

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

   3. Taylor expanded in n around -inf 26.2%

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

      1. mul-1-neg 26.2%

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

      2. distribute-neg-frac2 26.2%

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

   5. Simplified 26.2%

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

   6. Taylor expanded in x around inf 81.0%

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

      1. sub-neg 81.0%

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

      2. associate-*r/ 81.0%

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

      3. metadata-eval 81.0%

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

      4. metadata-eval 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in x around inf 84.7%

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

       1. associate-*r/ 84.7%

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

       2. *-commutative 84.7%

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

       3. log-rec 84.7%

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

   11. Simplified 84.7%

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

   if 2.00000000000000007e-10 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000003e181

   1. Initial program 67.6%

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

   if 5.0000000000000003e181 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.5%

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

   3. Taylor expanded in x around inf 0.3%

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

      1. mul-1-neg 0.3%

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

      2. log-rec 0.3%

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

      3. mul-1-neg 0.3%

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

      4. distribute-neg-frac 0.3%

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

      5. mul-1-neg 0.3%

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

      6. remove-double-neg 0.3%

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

      7. *-commutative 0.3%

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

   5. Simplified 0.3%

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

   6. Taylor expanded in n around inf 84.6%

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

      1. *-commutative 84.6%

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

   8. Simplified 84.6%

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

      1. log1p-expm1-u 94.6%

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

      2. associate-/r* 94.6%

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

   10. Applied egg-rr 94.6%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{x}}{n}\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n} - \frac{\frac{0.5}{x} + -1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+181}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{x}}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{t\_1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{t\_0 - \frac{\frac{0.5}{x} + -1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_0 - \log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+181}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{x}}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ (/ (* (log x) -2.0) x) n))) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-160)
     (/ (/ t_1 n) x)
     (if (<= (/ 1.0 n) 2e-54)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 1e-11)
         (/ (- t_0 (/ (+ (/ 0.5 x) -1.0) x)) n)
         (if (<= (/ 1.0 n) 2e-10)
           (/ (- t_0 (log (/ x (+ 1.0 x)))) n)
           (if (<= (/ 1.0 n) 5e+181)
             (- (pow (+ 1.0 x) (/ 1.0 n)) t_1)
             (log1p (expm1 (/ (/ 1.0 x) n))))))))))

C

double code(double x, double n) {
	double t_0 = -0.5 * (((log(x) * -2.0) / x) / n);
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = (t_0 - (((0.5 / x) + -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = (t_0 - log((x / (1.0 + x)))) / n;
	} else if ((1.0 / n) <= 5e+181) {
		tmp = pow((1.0 + x), (1.0 / n)) - t_1;
	} else {
		tmp = log1p(expm1(((1.0 / x) / n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = -0.5 * (((Math.log(x) * -2.0) / x) / n);
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = (t_0 - (((0.5 / x) + -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = (t_0 - Math.log((x / (1.0 + x)))) / n;
	} else if ((1.0 / n) <= 5e+181) {
		tmp = Math.pow((1.0 + x), (1.0 / n)) - t_1;
	} else {
		tmp = Math.log1p(Math.expm1(((1.0 / x) / n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -0.5 * (((math.log(x) * -2.0) / x) / n)
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-160:
		tmp = (t_1 / n) / x
	elif (1.0 / n) <= 2e-54:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 1e-11:
		tmp = (t_0 - (((0.5 / x) + -1.0) / x)) / n
	elif (1.0 / n) <= 2e-10:
		tmp = (t_0 - math.log((x / (1.0 + x)))) / n
	elif (1.0 / n) <= 5e+181:
		tmp = math.pow((1.0 + x), (1.0 / n)) - t_1
	else:
		tmp = math.log1p(math.expm1(((1.0 / x) / n)))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(-0.5 * Float64(Float64(Float64(log(x) * -2.0) / x) / n))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-160)
		tmp = Float64(Float64(t_1 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-54)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(0.5 / x) + -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e-10)
		tmp = Float64(Float64(t_0 - log(Float64(x / Float64(1.0 + x)))) / n);
	elseif (Float64(1.0 / n) <= 5e+181)
		tmp = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - t_1);
	else
		tmp = log1p(expm1(Float64(Float64(1.0 / x) / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(-0.5 * N[(N[(N[(N[Log[x], $MachinePrecision] * -2.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-160], N[(N[(t$95$1 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-54], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[(t$95$0 - N[(N[(N[(0.5 / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-10], N[(N[(t$95$0 - N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+181], N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision], N[Log[1 + N[(Exp[N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999999e-161

   1. Initial program 74.0%

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

   3. Taylor expanded in x around inf 88.5%

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

      1. associate-/r* 88.5%

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

      2. mul-1-neg 88.5%

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

      3. log-rec 88.5%

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

      4. mul-1-neg 88.5%

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

      5. distribute-neg-frac 88.5%

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

      6. mul-1-neg 88.5%

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

      7. remove-double-neg 88.5%

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

      8. *-rgt-identity 88.5%

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

      9. associate-/l* 88.5%

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

      10. exp-to-pow 88.5%

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

   5. Simplified 88.5%

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

   if -9.9999999999999999e-161 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-54

   1. Initial program 33.7%

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

   3. Taylor expanded in n around inf 90.5%

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

      1. log1p-define 90.5%

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

   5. Simplified 90.5%

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

      1. log1p-undefine 90.5%

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

      2. +-commutative 90.5%

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

      3. diff-log 90.5%

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

   7. Applied egg-rr 90.5%

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

   if 2.0000000000000001e-54 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999939e-12

   1. Initial program 4.8%

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

   3. Taylor expanded in n around -inf 26.2%

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

      1. mul-1-neg 26.2%

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

      2. distribute-neg-frac2 26.2%

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

   5. Simplified 26.2%

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

   6. Taylor expanded in x around inf 81.0%

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

      1. sub-neg 81.0%

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

      2. associate-*r/ 81.0%

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

      3. metadata-eval 81.0%

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

      4. metadata-eval 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in x around inf 84.7%

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

       1. associate-*r/ 84.7%

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

       2. *-commutative 84.7%

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

       3. log-rec 84.7%

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

   11. Simplified 84.7%

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

   if 9.99999999999999939e-12 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000007e-10

   1. Initial program 33.8%

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

   3. Taylor expanded in n around -inf 82.2%

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

      1. mul-1-neg 82.2%

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

      2. distribute-neg-frac2 82.2%

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

   5. Simplified 82.9%

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

      1. log1p-undefine 82.9%

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

      2. diff-log 88.7%

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

   7. Applied egg-rr 88.7%

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

      1. +-commutative 88.7%

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

   9. Simplified 88.7%

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

   10. Taylor expanded in x around inf 80.9%

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

       1. associate-*r/ 47.3%

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

       2. *-commutative 47.3%

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

       3. log-rec 47.3%

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

   12. Simplified 80.9%

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

   if 2.00000000000000007e-10 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000003e181

   1. Initial program 67.6%

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

   if 5.0000000000000003e181 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.5%

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

   3. Taylor expanded in x around inf 0.3%

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

      1. mul-1-neg 0.3%

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

      2. log-rec 0.3%

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

      3. mul-1-neg 0.3%

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

      4. distribute-neg-frac 0.3%

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

      5. mul-1-neg 0.3%

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

      6. remove-double-neg 0.3%

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

      7. *-commutative 0.3%

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

   5. Simplified 0.3%

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

   6. Taylor expanded in n around inf 84.6%

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

      1. *-commutative 84.6%

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

   8. Simplified 84.6%

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

      1. log1p-expm1-u 94.6%

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

      2. associate-/r* 94.6%

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

   10. Applied egg-rr 94.6%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{x}}{n}\right)\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n} - \frac{\frac{0.5}{x} + -1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n} - \log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+181}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{x}}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n} - \frac{\frac{0.5}{x} + -1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+181}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{x}}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -1e-160)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-54)
       t_1
       (if (<= (/ 1.0 n) 1e-11)
         (/
          (- (* -0.5 (/ (/ (* (log x) -2.0) x) n)) (/ (+ (/ 0.5 x) -1.0) x))
          n)
         (if (<= (/ 1.0 n) 2e-10)
           t_1
           (if (<= (/ 1.0 n) 5e+181)
             (- (+ 1.0 (/ x n)) t_0)
             (log1p (expm1 (/ (/ 1.0 x) n))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = ((-0.5 * (((log(x) * -2.0) / x) / n)) - (((0.5 / x) + -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+181) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = log1p(expm1(((1.0 / x) / n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = ((-0.5 * (((Math.log(x) * -2.0) / x) / n)) - (((0.5 / x) + -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+181) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.log1p(Math.expm1(((1.0 / x) / n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if (1.0 / n) <= -1e-160:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-54:
		tmp = t_1
	elif (1.0 / n) <= 1e-11:
		tmp = ((-0.5 * (((math.log(x) * -2.0) / x) / n)) - (((0.5 / x) + -1.0) / x)) / n
	elif (1.0 / n) <= 2e-10:
		tmp = t_1
	elif (1.0 / n) <= 5e+181:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.log1p(math.expm1(((1.0 / x) / n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-160)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-54)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(Float64(Float64(-0.5 * Float64(Float64(Float64(log(x) * -2.0) / x) / n)) - Float64(Float64(Float64(0.5 / x) + -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e-10)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e+181)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = log1p(expm1(Float64(Float64(1.0 / x) / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-160], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-54], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[(N[(-0.5 * N[(N[(N[(N[Log[x], $MachinePrecision] * -2.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.5 / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-10], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+181], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Log[1 + N[(Exp[N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999999e-161

   1. Initial program 74.0%

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

   3. Taylor expanded in x around inf 88.5%

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

      1. associate-/r* 88.5%

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

      2. mul-1-neg 88.5%

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

      3. log-rec 88.5%

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

      4. mul-1-neg 88.5%

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

      5. distribute-neg-frac 88.5%

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

      6. mul-1-neg 88.5%

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

      7. remove-double-neg 88.5%

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

      8. *-rgt-identity 88.5%

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

      9. associate-/l* 88.5%

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

      10. exp-to-pow 88.5%

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

   5. Simplified 88.5%

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

   if -9.9999999999999999e-161 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-54 or 9.99999999999999939e-12 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000007e-10

   1. Initial program 33.7%

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

   3. Taylor expanded in n around inf 90.2%

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

      1. log1p-define 90.2%

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

   5. Simplified 90.2%

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

      1. log1p-undefine 90.2%

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

      2. +-commutative 90.2%

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

      3. diff-log 90.2%

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

   7. Applied egg-rr 90.2%

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

   if 2.0000000000000001e-54 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999939e-12

   1. Initial program 4.8%

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

   3. Taylor expanded in n around -inf 26.2%

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

      1. mul-1-neg 26.2%

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

      2. distribute-neg-frac2 26.2%

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

   5. Simplified 26.2%

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

   6. Taylor expanded in x around inf 81.0%

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

      1. sub-neg 81.0%

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

      2. associate-*r/ 81.0%

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

      3. metadata-eval 81.0%

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

      4. metadata-eval 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in x around inf 84.7%

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

       1. associate-*r/ 84.7%

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

       2. *-commutative 84.7%

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

       3. log-rec 84.7%

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

   11. Simplified 84.7%

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

   if 2.00000000000000007e-10 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000003e181

   1. Initial program 67.6%

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

   3. Taylor expanded in x around 0 63.2%

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

      1. *-rgt-identity 63.2%

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

      2. associate-/l* 63.2%

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

      3. exp-to-pow 63.2%

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

   5. Simplified 63.2%

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

   if 5.0000000000000003e181 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.5%

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

   3. Taylor expanded in x around inf 0.3%

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

      1. mul-1-neg 0.3%

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

      2. log-rec 0.3%

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

      3. mul-1-neg 0.3%

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

      4. distribute-neg-frac 0.3%

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

      5. mul-1-neg 0.3%

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

      6. remove-double-neg 0.3%

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

      7. *-commutative 0.3%

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

   5. Simplified 0.3%

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

   6. Taylor expanded in n around inf 84.6%

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

      1. *-commutative 84.6%

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

   8. Simplified 84.6%

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

      1. log1p-expm1-u 94.6%

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

      2. associate-/r* 94.6%

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

   10. Applied egg-rr 94.6%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{x}}{n}\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n} - \frac{\frac{0.5}{x} + -1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+181}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{x}}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n} - \frac{\frac{0.5}{x} + -1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+181}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -1e-160)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-54)
       t_1
       (if (<= (/ 1.0 n) 1e-11)
         (/
          (- (* -0.5 (/ (/ (* (log x) -2.0) x) n)) (/ (+ (/ 0.5 x) -1.0) x))
          n)
         (if (<= (/ 1.0 n) 2e-10)
           t_1
           (if (<= (/ 1.0 n) 5e+181)
             (- (+ 1.0 (/ x n)) t_0)
             (/ (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) x) n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = ((-0.5 * (((log(x) * -2.0) / x) / n)) - (((0.5 / x) + -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+181) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = log(((1.0d0 + x) / x)) / n
    if ((1.0d0 / n) <= (-1d-160)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 2d-54) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d-11) then
        tmp = (((-0.5d0) * (((log(x) * (-2.0d0)) / x) / n)) - (((0.5d0 / x) + (-1.0d0)) / x)) / n
    else if ((1.0d0 / n) <= 2d-10) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d+181) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = ((1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-54) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = ((-0.5 * (((Math.log(x) * -2.0) / x) / n)) - (((0.5 / x) + -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+181) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if (1.0 / n) <= -1e-160:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-54:
		tmp = t_1
	elif (1.0 / n) <= 1e-11:
		tmp = ((-0.5 * (((math.log(x) * -2.0) / x) / n)) - (((0.5 / x) + -1.0) / x)) / n
	elif (1.0 / n) <= 2e-10:
		tmp = t_1
	elif (1.0 / n) <= 5e+181:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-160)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-54)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(Float64(Float64(-0.5 * Float64(Float64(Float64(log(x) * -2.0) / x) / n)) - Float64(Float64(Float64(0.5 / x) + -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e-10)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e+181)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log(((1.0 + x) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -1e-160)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 2e-54)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e-11)
		tmp = ((-0.5 * (((log(x) * -2.0) / x) / n)) - (((0.5 / x) + -1.0) / x)) / n;
	elseif ((1.0 / n) <= 2e-10)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e+181)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-160], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-54], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[(N[(-0.5 * N[(N[(N[(N[Log[x], $MachinePrecision] * -2.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.5 / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-10], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+181], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999999e-161

   1. Initial program 74.0%

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

   3. Taylor expanded in x around inf 88.5%

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

      1. associate-/r* 88.5%

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

      2. mul-1-neg 88.5%

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

      3. log-rec 88.5%

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

      4. mul-1-neg 88.5%

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

      5. distribute-neg-frac 88.5%

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

      6. mul-1-neg 88.5%

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

      7. remove-double-neg 88.5%

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

      8. *-rgt-identity 88.5%

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

      9. associate-/l* 88.5%

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

      10. exp-to-pow 88.5%

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

   5. Simplified 88.5%

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

   if -9.9999999999999999e-161 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-54 or 9.99999999999999939e-12 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000007e-10

   1. Initial program 33.7%

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

   3. Taylor expanded in n around inf 90.2%

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

      1. log1p-define 90.2%

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

   5. Simplified 90.2%

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

      1. log1p-undefine 90.2%

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

      2. +-commutative 90.2%

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

      3. diff-log 90.2%

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

   7. Applied egg-rr 90.2%

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

   if 2.0000000000000001e-54 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999939e-12

   1. Initial program 4.8%

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

   3. Taylor expanded in n around -inf 26.2%

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

      1. mul-1-neg 26.2%

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

      2. distribute-neg-frac2 26.2%

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

   5. Simplified 26.2%

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

   6. Taylor expanded in x around inf 81.0%

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

      1. sub-neg 81.0%

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

      2. associate-*r/ 81.0%

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

      3. metadata-eval 81.0%

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

      4. metadata-eval 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in x around inf 84.7%

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

       1. associate-*r/ 84.7%

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

       2. *-commutative 84.7%

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

       3. log-rec 84.7%

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

   11. Simplified 84.7%

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

   if 2.00000000000000007e-10 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000003e181

   1. Initial program 67.6%

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

   3. Taylor expanded in x around 0 63.2%

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

      1. *-rgt-identity 63.2%

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

      2. associate-/l* 63.2%

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

      3. exp-to-pow 63.2%

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

   5. Simplified 63.2%

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

   if 5.0000000000000003e181 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.5%

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

   3. Taylor expanded in n around inf 8.3%

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

      1. log1p-define 8.3%

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

   5. Simplified 8.3%

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

   6. Taylor expanded in x around inf 94.6%

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

   8. Simplified 94.6%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{\log x \cdot -2}{x}}{n} - \frac{\frac{0.5}{x} + -1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+181}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}{n}\\ t_2 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+179}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+181}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n))))
        (t_1 (/ (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) x) n))
        (t_2 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -2e+179)
     t_2
     (if (<= (/ 1.0 n) -2e-6)
       t_0
       (if (<= (/ 1.0 n) -1e-160)
         t_1
         (if (<= (/ 1.0 n) 2e-10) t_2 (if (<= (/ 1.0 n) 5e+181) t_0 t_1)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	double t_2 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -2e+179) {
		tmp = t_2;
	} else if ((1.0 / n) <= -2e-6) {
		tmp = t_0;
	} else if ((1.0 / n) <= -1e-160) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = t_2;
	} else if ((1.0 / n) <= 5e+181) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = ((1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / x) / n
    t_2 = log(((1.0d0 + x) / x)) / n
    if ((1.0d0 / n) <= (-2d+179)) then
        tmp = t_2
    else if ((1.0d0 / n) <= (-2d-6)) then
        tmp = t_0
    else if ((1.0d0 / n) <= (-1d-160)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-10) then
        tmp = t_2
    else if ((1.0d0 / n) <= 5d+181) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double t_1 = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	double t_2 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -2e+179) {
		tmp = t_2;
	} else if ((1.0 / n) <= -2e-6) {
		tmp = t_0;
	} else if ((1.0 / n) <= -1e-160) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = t_2;
	} else if ((1.0 / n) <= 5e+181) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n
	t_2 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if (1.0 / n) <= -2e+179:
		tmp = t_2
	elif (1.0 / n) <= -2e-6:
		tmp = t_0
	elif (1.0 / n) <= -1e-160:
		tmp = t_1
	elif (1.0 / n) <= 2e-10:
		tmp = t_2
	elif (1.0 / n) <= 5e+181:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / x) / n)
	t_2 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+179)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= -2e-6)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -1e-160)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-10)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 5e+181)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	t_2 = log(((1.0 + x) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -2e+179)
		tmp = t_2;
	elseif ((1.0 / n) <= -2e-6)
		tmp = t_0;
	elseif ((1.0 / n) <= -1e-160)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-10)
		tmp = t_2;
	elseif ((1.0 / n) <= 5e+181)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+179], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-6], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-160], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-10], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+181], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999996e179 or -9.9999999999999999e-161 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000007e-10

   1. Initial program 49.5%

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

   3. Taylor expanded in n around inf 78.2%

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

      1. log1p-define 78.2%

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

   5. Simplified 78.2%

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

      1. log1p-undefine 78.2%

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

      2. +-commutative 78.2%

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

      3. diff-log 78.3%

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

   7. Applied egg-rr 78.3%

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

   if -1.99999999999999996e179 < (/.f64 #s(literal 1 binary64) n) < -1.99999999999999991e-6 or 2.00000000000000007e-10 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000003e181

   1. Initial program 85.4%

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

   3. Taylor expanded in x around 0 58.3%

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

      1. *-rgt-identity 58.3%

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

      2. associate-/l* 58.3%

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

      3. exp-to-pow 58.3%

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

   5. Simplified 58.3%

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

   if -1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) n) < -9.9999999999999999e-161 or 5.0000000000000003e181 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 10.4%

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

   3. Taylor expanded in n around inf 34.1%

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

      1. log1p-define 34.1%

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

   5. Simplified 34.1%

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

   6. Taylor expanded in x around inf 75.3%

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

   8. Simplified 75.3%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+179}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+181}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+181}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-160)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-10)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 5e+181)
         (- (+ 1.0 (/ x n)) t_0)
         (/ (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) x) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+181) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1d-160)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 2d-10) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 5d+181) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = ((1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+181) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-160:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-10:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 5e+181:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-160)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-10)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+181)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-160)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 2e-10)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 5e+181)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-160], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-10], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+181], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999999e-161

   1. Initial program 74.0%

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

   3. Taylor expanded in x around inf 88.5%

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

      1. associate-/r* 88.5%

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

      2. mul-1-neg 88.5%

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

      3. log-rec 88.5%

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

      4. mul-1-neg 88.5%

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

      5. distribute-neg-frac 88.5%

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

      6. mul-1-neg 88.5%

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

      7. remove-double-neg 88.5%

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

      8. *-rgt-identity 88.5%

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

      9. associate-/l* 88.5%

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

      10. exp-to-pow 88.5%

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

   5. Simplified 88.5%

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

   if -9.9999999999999999e-161 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000007e-10

   1. Initial program 31.7%

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

   3. Taylor expanded in n around inf 85.7%

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

      1. log1p-define 85.7%

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

   5. Simplified 85.7%

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

      1. log1p-undefine 85.7%

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

      2. +-commutative 85.7%

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

      3. diff-log 85.7%

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

   7. Applied egg-rr 85.7%

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

   if 2.00000000000000007e-10 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000003e181

   1. Initial program 67.6%

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

   3. Taylor expanded in x around 0 63.2%

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

      1. *-rgt-identity 63.2%

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

      2. associate-/l* 63.2%

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

      3. exp-to-pow 63.2%

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

   5. Simplified 63.2%

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

   if 5.0000000000000003e181 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.5%

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

   3. Taylor expanded in n around inf 8.3%

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

      1. log1p-define 8.3%

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

   5. Simplified 8.3%

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

   6. Taylor expanded in x around inf 94.6%

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

   8. Simplified 94.6%

      \[\leadsto \frac{\color{blue}{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+181}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 80.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+181}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-160)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-10)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 5e+181)
         (- 1.0 t_0)
         (/ (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) x) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+181) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1d-160)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 2d-10) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 5d+181) then
        tmp = 1.0d0 - t_0
    else
        tmp = ((1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-160) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+181) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-160:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-10:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 5e+181:
		tmp = 1.0 - t_0
	else:
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-160)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-10)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+181)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-160)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 2e-10)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 5e+181)
		tmp = 1.0 - t_0;
	else
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-160], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-10], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+181], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999999e-161

   1. Initial program 74.0%

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

   3. Taylor expanded in x around inf 88.5%

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

      1. associate-/r* 88.5%

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

      2. mul-1-neg 88.5%

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

      3. log-rec 88.5%

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

      4. mul-1-neg 88.5%

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

      5. distribute-neg-frac 88.5%

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

      6. mul-1-neg 88.5%

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

      7. remove-double-neg 88.5%

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

      8. *-rgt-identity 88.5%

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

      9. associate-/l* 88.5%

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

      10. exp-to-pow 88.5%

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

   5. Simplified 88.5%

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

   if -9.9999999999999999e-161 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000007e-10

   1. Initial program 31.7%

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

   3. Taylor expanded in n around inf 85.7%

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

      1. log1p-define 85.7%

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

   5. Simplified 85.7%

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

      1. log1p-undefine 85.7%

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

      2. +-commutative 85.7%

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

      3. diff-log 85.7%

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

   7. Applied egg-rr 85.7%

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

   if 2.00000000000000007e-10 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000003e181

   1. Initial program 67.6%

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

   3. Taylor expanded in x around 0 60.6%

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

      1. *-rgt-identity 60.6%

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

      2. associate-/l* 60.6%

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

      3. exp-to-pow 60.6%

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

   5. Simplified 60.6%

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

   if 5.0000000000000003e181 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.5%

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

   3. Taylor expanded in n around inf 8.3%

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

      1. log1p-define 8.3%

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

   5. Simplified 8.3%

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

   6. Taylor expanded in x around inf 94.6%

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

   8. Simplified 94.6%

      \[\leadsto \frac{\color{blue}{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+181}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{-220}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-181}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+178}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{n \cdot {x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.32e-220)
   (/ (log x) (- n))
   (if (<= x 5.2e-181)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 0.88)
       (/ (- x (log x)) n)
       (if (<= x 1.2e+178)
         (/
          (/ (+ 1.0 (/ (- -0.5 (/ (+ -0.3333333333333333 (/ 0.25 x)) x)) x)) x)
          n)
         (/ -0.5 (* n (pow x 2.0))))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.32e-220) {
		tmp = log(x) / -n;
	} else if (x <= 5.2e-181) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.88) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.2e+178) {
		tmp = ((1.0 + ((-0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = -0.5 / (n * pow(x, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.32d-220) then
        tmp = log(x) / -n
    else if (x <= 5.2d-181) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.88d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.2d+178) then
        tmp = ((1.0d0 + (((-0.5d0) - (((-0.3333333333333333d0) + (0.25d0 / x)) / x)) / x)) / x) / n
    else
        tmp = (-0.5d0) / (n * (x ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.32e-220) {
		tmp = Math.log(x) / -n;
	} else if (x <= 5.2e-181) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.88) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.2e+178) {
		tmp = ((1.0 + ((-0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = -0.5 / (n * Math.pow(x, 2.0));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.32e-220:
		tmp = math.log(x) / -n
	elif x <= 5.2e-181:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.88:
		tmp = (x - math.log(x)) / n
	elif x <= 1.2e+178:
		tmp = ((1.0 + ((-0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / x)) / x) / n
	else:
		tmp = -0.5 / (n * math.pow(x, 2.0))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.32e-220)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 5.2e-181)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.88)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.2e+178)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 - Float64(Float64(-0.3333333333333333 + Float64(0.25 / x)) / x)) / x)) / x) / n);
	else
		tmp = Float64(-0.5 / Float64(n * (x ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.32e-220)
		tmp = log(x) / -n;
	elseif (x <= 5.2e-181)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.88)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.2e+178)
		tmp = ((1.0 + ((-0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / x)) / x) / n;
	else
		tmp = -0.5 / (n * (x ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.32e-220], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 5.2e-181], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.88], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.2e+178], N[(N[(N[(1.0 + N[(N[(-0.5 - N[(N[(-0.3333333333333333 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(-0.5 / N[(n * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.31999999999999996e-220

   1. Initial program 42.4%

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

   3. Taylor expanded in x around 0 42.4%

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

      1. *-rgt-identity 42.4%

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

      2. associate-/l* 42.4%

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

      3. exp-to-pow 42.4%

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

   5. Simplified 42.4%

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

   6. Taylor expanded in n around inf 57.0%

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

      1. associate-*r/ 57.0%

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

      2. neg-mul-1 57.0%

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

   8. Simplified 57.0%

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

   if 1.31999999999999996e-220 < x < 5.19999999999999998e-181

   1. Initial program 61.5%

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

   3. Taylor expanded in x around 0 61.5%

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

      1. *-rgt-identity 61.5%

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

      2. associate-/l* 61.5%

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

      3. exp-to-pow 61.5%

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

   5. Simplified 61.5%

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

   if 5.19999999999999998e-181 < x < 0.880000000000000004

   1. Initial program 36.5%

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

   3. Taylor expanded in n around inf 48.8%

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

      1. log1p-define 48.8%

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

   5. Simplified 48.8%

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

   6. Taylor expanded in x around 0 47.0%

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

   if 0.880000000000000004 < x < 1.2e178

   1. Initial program 53.9%

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

   3. Taylor expanded in n around inf 52.3%

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

      1. log1p-define 52.3%

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

   5. Simplified 52.3%

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

   6. Taylor expanded in x around inf 59.6%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{0.3333333333333333}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{x} + 0.25 \cdot \frac{1}{{x}^{3}}\right)}{x}}}{n} \]

   8. Simplified 59.6%

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

   if 1.2e178 < x

   1. Initial program 87.3%

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

   3. Taylor expanded in n around -inf 87.3%

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

      1. mul-1-neg 87.3%

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

      2. distribute-neg-frac2 87.3%

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

   5. Simplified 87.3%

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

   6. Taylor expanded in x around inf 62.0%

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

      1. sub-neg 62.0%

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

      2. associate-*r/ 62.0%

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

      3. metadata-eval 62.0%

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

      4. metadata-eval 62.0%

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

   8. Simplified 62.0%

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

   9. Taylor expanded in x around 0 87.3%

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

       1. *-commutative 87.3%

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

   11. Simplified 87.3%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{-220}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-181}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+178}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{n \cdot {x}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.9 \cdot 10^{-201}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-181}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.9e-201)
   (/ (log x) (- n))
   (if (<= x 7.5e-181)
     (/ 1.0 (* n x))
     (if (<= x 0.9)
       (/ (- x (log x)) n)
       (/
        (/ (+ 1.0 (/ (- -0.5 (/ (+ -0.3333333333333333 (/ 0.25 x)) x)) x)) x)
        n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.9e-201) {
		tmp = log(x) / -n;
	} else if (x <= 7.5e-181) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.9) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 + ((-0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 4.9d-201) then
        tmp = log(x) / -n
    else if (x <= 7.5d-181) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 0.9d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 + (((-0.5d0) - (((-0.3333333333333333d0) + (0.25d0 / x)) / x)) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.9e-201) {
		tmp = Math.log(x) / -n;
	} else if (x <= 7.5e-181) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.9) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 + ((-0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.9e-201:
		tmp = math.log(x) / -n
	elif x <= 7.5e-181:
		tmp = 1.0 / (n * x)
	elif x <= 0.9:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 + ((-0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.9e-201)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 7.5e-181)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 0.9)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 - Float64(Float64(-0.3333333333333333 + Float64(0.25 / x)) / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.9e-201)
		tmp = log(x) / -n;
	elseif (x <= 7.5e-181)
		tmp = 1.0 / (n * x);
	elseif (x <= 0.9)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 + ((-0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.9e-201], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 7.5e-181], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.9], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(-0.5 - N[(N[(-0.3333333333333333 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 4.8999999999999997e-201

   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 x around 0 44.4%

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

      1. *-rgt-identity 44.4%

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

      2. associate-/l* 44.4%

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

      3. exp-to-pow 44.4%

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

   5. Simplified 44.4%

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

   6. Taylor expanded in n around inf 55.9%

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

      1. associate-*r/ 55.9%

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

      2. neg-mul-1 55.9%

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

   8. Simplified 55.9%

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

   if 4.8999999999999997e-201 < x < 7.5000000000000002e-181

   1. Initial program 61.4%

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

   3. Taylor expanded in x around inf 43.9%

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

      1. mul-1-neg 43.9%

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

      2. log-rec 43.9%

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

      3. mul-1-neg 43.9%

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

      4. distribute-neg-frac 43.9%

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

      5. mul-1-neg 43.9%

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

      6. remove-double-neg 43.9%

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

      7. *-commutative 43.9%

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

   5. Simplified 43.9%

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

   6. Taylor expanded in n around inf 65.6%

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

      1. *-commutative 65.6%

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

   8. Simplified 65.6%

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

   if 7.5000000000000002e-181 < x < 0.900000000000000022

   1. Initial program 36.5%

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

   3. Taylor expanded in n around inf 48.8%

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

      1. log1p-define 48.8%

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

   5. Simplified 48.8%

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

   6. Taylor expanded in x around 0 47.0%

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

   if 0.900000000000000022 < x

   1. Initial program 66.8%

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

   3. Taylor expanded in n around inf 65.8%

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

      1. log1p-define 65.8%

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

   5. Simplified 65.8%

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

   6. Taylor expanded in x around inf 60.5%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{0.3333333333333333}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{x} + 0.25 \cdot \frac{1}{{x}^{3}}\right)}{x}}}{n} \]

   8. Simplified 60.5%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{x}}{x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.9 \cdot 10^{-201}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-181}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 56.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{-220}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-181}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.15e-220)
   (/ (log x) (- n))
   (if (<= x 2.4e-181)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 0.9)
       (/ (- x (log x)) n)
       (/
        (/ (+ 1.0 (/ (- -0.5 (/ (+ -0.3333333333333333 (/ 0.25 x)) x)) x)) x)
        n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.15e-220) {
		tmp = log(x) / -n;
	} else if (x <= 2.4e-181) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.9) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 + ((-0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 2.15d-220) then
        tmp = log(x) / -n
    else if (x <= 2.4d-181) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.9d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 + (((-0.5d0) - (((-0.3333333333333333d0) + (0.25d0 / x)) / x)) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.15e-220) {
		tmp = Math.log(x) / -n;
	} else if (x <= 2.4e-181) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.9) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 + ((-0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.15e-220:
		tmp = math.log(x) / -n
	elif x <= 2.4e-181:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.9:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 + ((-0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.15e-220)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 2.4e-181)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.9)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 - Float64(Float64(-0.3333333333333333 + Float64(0.25 / x)) / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.15e-220)
		tmp = log(x) / -n;
	elseif (x <= 2.4e-181)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.9)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 + ((-0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.15e-220], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 2.4e-181], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.9], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(-0.5 - N[(N[(-0.3333333333333333 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.1499999999999999e-220

   1. Initial program 42.4%

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

   3. Taylor expanded in x around 0 42.4%

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

      1. *-rgt-identity 42.4%

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

      2. associate-/l* 42.4%

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

      3. exp-to-pow 42.4%

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

   5. Simplified 42.4%

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

   6. Taylor expanded in n around inf 57.0%

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

      1. associate-*r/ 57.0%

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

      2. neg-mul-1 57.0%

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

   8. Simplified 57.0%

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

   if 2.1499999999999999e-220 < x < 2.4000000000000001e-181

   1. Initial program 61.5%

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

   3. Taylor expanded in x around 0 61.5%

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

      1. *-rgt-identity 61.5%

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

      2. associate-/l* 61.5%

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

      3. exp-to-pow 61.5%

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

   5. Simplified 61.5%

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

   if 2.4000000000000001e-181 < x < 0.900000000000000022

   1. Initial program 36.5%

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

   3. Taylor expanded in n around inf 48.8%

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

      1. log1p-define 48.8%

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

   5. Simplified 48.8%

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

   6. Taylor expanded in x around 0 47.0%

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

   if 0.900000000000000022 < x

   1. Initial program 66.8%

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

   3. Taylor expanded in n around inf 65.8%

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

      1. log1p-define 65.8%

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

   5. Simplified 65.8%

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

   6. Taylor expanded in x around inf 60.5%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{0.3333333333333333}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{x} + 0.25 \cdot \frac{1}{{x}^{3}}\right)}{x}}}{n} \]

   8. Simplified 60.5%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{x}}{x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{-220}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-181}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 52.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 1.95 \cdot 10^{-199}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-181}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-59}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 1.95e-199)
     t_0
     (if (<= x 2.25e-181)
       (/ 1.0 (* n x))
       (if (<= x 4.5e-59)
         t_0
         (/ (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) x) n))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 1.95e-199) {
		tmp = t_0;
	} else if (x <= 2.25e-181) {
		tmp = 1.0 / (n * x);
	} else if (x <= 4.5e-59) {
		tmp = t_0;
	} else {
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 1.95d-199) then
        tmp = t_0
    else if (x <= 2.25d-181) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 4.5d-59) then
        tmp = t_0
    else
        tmp = ((1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 1.95e-199) {
		tmp = t_0;
	} else if (x <= 2.25e-181) {
		tmp = 1.0 / (n * x);
	} else if (x <= 4.5e-59) {
		tmp = t_0;
	} else {
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 1.95e-199:
		tmp = t_0
	elif x <= 2.25e-181:
		tmp = 1.0 / (n * x)
	elif x <= 4.5e-59:
		tmp = t_0
	else:
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 1.95e-199)
		tmp = t_0;
	elseif (x <= 2.25e-181)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 4.5e-59)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 1.95e-199)
		tmp = t_0;
	elseif (x <= 2.25e-181)
		tmp = 1.0 / (n * x);
	elseif (x <= 4.5e-59)
		tmp = t_0;
	else
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 1.95e-199], t$95$0, If[LessEqual[x, 2.25e-181], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.5e-59], t$95$0, N[(N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.9500000000000001e-199 or 2.2499999999999999e-181 < x < 4.50000000000000012e-59

   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 x around 0 38.7%

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

      1. *-rgt-identity 38.7%

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

      2. associate-/l* 38.7%

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

      3. exp-to-pow 38.7%

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

   5. Simplified 38.7%

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

   6. Taylor expanded in n around inf 55.5%

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

      1. associate-*r/ 55.5%

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

      2. neg-mul-1 55.5%

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

   8. Simplified 55.5%

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

   if 1.9500000000000001e-199 < x < 2.2499999999999999e-181

   1. Initial program 61.4%

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

   3. Taylor expanded in x around inf 43.9%

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

      1. mul-1-neg 43.9%

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

      2. log-rec 43.9%

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

      3. mul-1-neg 43.9%

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

      4. distribute-neg-frac 43.9%

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

      5. mul-1-neg 43.9%

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

      6. remove-double-neg 43.9%

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

      7. *-commutative 43.9%

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

   5. Simplified 43.9%

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

   6. Taylor expanded in n around inf 65.6%

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

      1. *-commutative 65.6%

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

   8. Simplified 65.6%

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

   if 4.50000000000000012e-59 < x

   1. Initial program 61.6%

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

   3. Taylor expanded in n around inf 59.5%

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

      1. log1p-define 59.5%

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

   5. Simplified 59.5%

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

   6. Taylor expanded in x around inf 54.2%

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

   8. Simplified 54.2%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{-199}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-181}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 46.2% accurate, 12.4× speedup

Math

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

FPCore

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

C

double code(double x, double n) {
	return ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / x)) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 / n) + ((((-0.5d0) / n) + (0.3333333333333333d0 / (n * x))) / x)) / x
end function

Java

public static double code(double x, double n) {
	return ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / x)) / x;
}

Python

def code(x, n):
	return ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / x)) / x

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(0.3333333333333333 / Float64(n * x))) / x)) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / x)) / x;
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 52.0%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 55.7%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-define 55.7%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

5. Simplified 55.7%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

6. Taylor expanded in x around -inf 45.3%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
7. Step-by-step derivation

   1. associate-*r/ 45.3%

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}\right)}{x}} \]

8. Simplified 45.3%

   \[\leadsto \color{blue}{\frac{\frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x} + \frac{1}{n}}{x}} \]

9. Final simplification 45.3%

   \[\leadsto \frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{n \cdot x}}{x}}{x} \]
10. Add Preprocessing

Alternative 21: 46.2% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}{n} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) x) n))

C

double code(double x, double n) {
	return ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / x) / n
end function

Java

public static double code(double x, double n) {
	return ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
}

Python

def code(x, n):
	return ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / x) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]

Derivation

1. Initial program 52.0%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 55.7%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-define 55.7%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

5. Simplified 55.7%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

6. Taylor expanded in x around inf 45.3%

   \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{0.3333333333333333}{{x}^{2}}\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]
7. Step-by-step derivation

8. Simplified 45.3%

   \[\leadsto \frac{\color{blue}{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}}{n} \]

9. Final simplification 45.3%

   \[\leadsto \frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}{n} \]
10. Add Preprocessing

Alternative 22: 39.9% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{n \cdot x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ 1.0 (* n x)))

C

double code(double x, double n) {
	return 1.0 / (n * x);
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 1.0d0 / (n * x)
end function

Java

public static double code(double x, double n) {
	return 1.0 / (n * x);
}

Python

def code(x, n):
	return 1.0 / (n * x)

Julia

function code(x, n)
	return Float64(1.0 / Float64(n * x))
end

MATLAB

function tmp = code(x, n)
	tmp = 1.0 / (n * x);
end

Wolfram

code[x_, n_] := N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.0%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 56.2%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
4. Step-by-step derivation

   1. mul-1-neg 56.2%

      \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

   2. log-rec 56.2%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   3. mul-1-neg 56.2%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

   4. distribute-neg-frac 56.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

   5. mul-1-neg 56.2%

      \[\leadsto \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]

   6. remove-double-neg 56.2%

      \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

   7. *-commutative 56.2%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

5. Simplified 56.2%

   \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

6. Taylor expanded in n around inf 40.5%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
7. Step-by-step derivation

   1. *-commutative 40.5%

      \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

8. Simplified 40.5%

   \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]

9. Final simplification 40.5%

   \[\leadsto \frac{1}{n \cdot x} \]
10. Add Preprocessing

Alternative 23: 40.5% 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 52.0%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 56.2%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
4. Step-by-step derivation

   1. mul-1-neg 56.2%

      \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

   2. log-rec 56.2%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   3. mul-1-neg 56.2%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

   4. distribute-neg-frac 56.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

   5. mul-1-neg 56.2%

      \[\leadsto \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]

   6. remove-double-neg 56.2%

      \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

   7. *-commutative 56.2%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

5. Simplified 56.2%

   \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

6. Taylor expanded in n around inf 40.5%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
7. Step-by-step derivation

   1. *-commutative 40.5%

      \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

8. Simplified 40.5%

   \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
9. Step-by-step derivation

   1. associate-/r* 40.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{n}} \]

   2. div-inv 40.7%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1}{n}} \]

10. Applied egg-rr 40.7%

    \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1}{n}} \]

11. Taylor expanded in x around 0 40.5%

    \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
12. Step-by-step derivation

    1. associate-/r* 40.7%

       \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

13. Simplified 40.7%

    \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

14. Final simplification 40.7%

    \[\leadsto \frac{\frac{1}{n}}{x} \]
15. Add Preprocessing

Alternative 24: 40.5% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 x) n))

C

double code(double x, double n) {
	return (1.0 / x) / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 / x) / n
end function

Java

public static double code(double x, double n) {
	return (1.0 / x) / n;
}

Python

def code(x, n):
	return (1.0 / x) / n

Julia

function code(x, n)
	return Float64(Float64(1.0 / x) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / x) / n;
end

Wolfram

code[x_, n_] := N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]

Derivation

1. Initial program 52.0%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 55.7%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-define 55.7%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

5. Simplified 55.7%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

6. Taylor expanded in x around inf 40.7%

   \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

7. Final simplification 40.7%

   \[\leadsto \frac{\frac{1}{x}}{n} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024100 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))