2nthrt (problem 3.4.6)

Percentage Accurate: 53.7% → 85.1%

Time: 26.4s

Alternatives: 21

Speedup: 1.9×

• could not determine a ground truth

  1. x = -1.0839270543163638e-218
  2. n = -1.9737286553345202e-289

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-133}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{t\_0}{n} + t\_0 \cdot \left(\frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x} + \frac{\frac{0.16666666666666666}{{n}^{3}} + \left(\frac{0.3333333333333333}{n} + {n}^{-2} \cdot -0.5\right)}{{x}^{2}}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(e^{\frac{x}{n}} - t\_0\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-5)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 1e-133)
       (/
        (+
         (log1p x)
         (- (* 0.5 (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n)) (log x)))
        n)
       (if (<= (/ 1.0 n) 2e-7)
         (/
          (+
           (/ t_0 n)
           (*
            t_0
            (+
             (/ (fma 0.5 (pow n -2.0) (/ -0.5 n)) x)
             (/
              (+
               (/ 0.16666666666666666 (pow n 3.0))
               (+ (/ 0.3333333333333333 n) (* (pow n -2.0) -0.5)))
              (pow x 2.0)))))
          x)
         (pow (pow (- (exp (/ x n)) t_0) 3.0) 0.3333333333333333))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-133) {
		tmp = (log1p(x) + ((0.5 * ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n)) - log(x))) / n;
	} else if ((1.0 / n) <= 2e-7) {
		tmp = ((t_0 / n) + (t_0 * ((fma(0.5, pow(n, -2.0), (-0.5 / n)) / x) + (((0.16666666666666666 / pow(n, 3.0)) + ((0.3333333333333333 / n) + (pow(n, -2.0) * -0.5))) / pow(x, 2.0))))) / x;
	} else {
		tmp = pow(pow((exp((x / n)) - t_0), 3.0), 0.3333333333333333);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-5)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-133)
		tmp = Float64(Float64(log1p(x) + Float64(Float64(0.5 * Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / n)) - log(x))) / n);
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = Float64(Float64(Float64(t_0 / n) + Float64(t_0 * Float64(Float64(fma(0.5, (n ^ -2.0), Float64(-0.5 / n)) / x) + Float64(Float64(Float64(0.16666666666666666 / (n ^ 3.0)) + Float64(Float64(0.3333333333333333 / n) + Float64((n ^ -2.0) * -0.5))) / (x ^ 2.0))))) / x);
	else
		tmp = (Float64(exp(Float64(x / n)) - t_0) ^ 3.0) ^ 0.3333333333333333;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-5], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-133], N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], N[(N[(N[(t$95$0 / n), $MachinePrecision] + N[(t$95$0 * N[(N[(N[(0.5 * N[Power[n, -2.0], $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(0.16666666666666666 / N[Power[n, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 / n), $MachinePrecision] + N[(N[Power[n, -2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[Power[N[Power[N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 96.1%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. log-rec 99.9%

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

      2. mul-1-neg 99.9%

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

      3. neg-mul-1 99.9%

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

      4. mul-1-neg 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-rgt-identity 99.9%

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

      8. associate-/l* 99.9%

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

      9. exp-to-pow 99.9%

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

      10. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n) < 1.0000000000000001e-133

   1. Initial program 39.0%

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

   3. Taylor expanded in n around inf 83.7%

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

      1. associate--l+ 83.7%

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

      2. log1p-define 83.7%

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

      3. +-commutative 83.7%

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

      4. associate--r+ 83.7%

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

      5. distribute-lft-out-- 83.7%

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

      6. div-sub 83.7%

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

      7. log1p-define 83.7%

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

   5. Simplified 83.7%

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

   if 1.0000000000000001e-133 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e-7

   1. Initial program 15.9%

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

   3. Taylor expanded in x around inf 69.7%

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

   5. Simplified 69.7%

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

   if 1.9999999999999999e-7 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 47.1%

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

      1. add-cbrt-cube 47.1%

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

      2. pow3 47.1%

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

      3. pow-to-exp 47.1%

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

      4. un-div-inv 47.1%

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

      5. +-commutative 47.1%

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

      6. log1p-define 99.7%

         \[\leadsto \sqrt[3]{{\left(e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3}}} - {x}^{\left(\frac{1}{n}\right)} \]
   5. Step-by-step derivation

      1. rem-cbrt-cube 99.8%

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

      2. *-un-lft-identity 99.8%

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

      3. exp-prod 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. add-cbrt-cube 99.9%

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

      2. pow1/3 99.8%

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

      3. pow3 99.8%

         \[\leadsto {\color{blue}{\left({\left({\left(e^{1}\right)}^{\left(\frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. pow-exp 100.0%

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

      5. *-un-lft-identity 100.0%

         \[\leadsto {\left({\left(e^{\color{blue}{\frac{x}{n}}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 100.0%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-133}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n} + {x}^{\left(\frac{1}{n}\right)} \cdot \left(\frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x} + \frac{\frac{0.16666666666666666}{{n}^{3}} + \left(\frac{0.3333333333333333}{n} + {n}^{-2} \cdot -0.5\right)}{{x}^{2}}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1 + \frac{\log x}{n}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(e^{\frac{x}{n}} - t\_0\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-5)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-133)
       (/
        (+
         (log1p x)
         (- (* 0.5 (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n)) (log x)))
        n)
       (if (<= (/ 1.0 n) 2e-7)
         (/ (/ (+ 1.0 (/ (log x) n)) x) n)
         (pow (pow (- (exp (/ x n)) t_0) 3.0) 0.3333333333333333))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-133) {
		tmp = (log1p(x) + ((0.5 * ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n)) - log(x))) / n;
	} else if ((1.0 / n) <= 2e-7) {
		tmp = ((1.0 + (log(x) / n)) / x) / n;
	} else {
		tmp = pow(pow((exp((x / n)) - t_0), 3.0), 0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-133) {
		tmp = (Math.log1p(x) + ((0.5 * ((Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0)) / n)) - Math.log(x))) / n;
	} else if ((1.0 / n) <= 2e-7) {
		tmp = ((1.0 + (Math.log(x) / n)) / x) / n;
	} else {
		tmp = Math.pow(Math.pow((Math.exp((x / n)) - t_0), 3.0), 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-5:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-133:
		tmp = (math.log1p(x) + ((0.5 * ((math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0)) / n)) - math.log(x))) / n
	elif (1.0 / n) <= 2e-7:
		tmp = ((1.0 + (math.log(x) / n)) / x) / n
	else:
		tmp = math.pow(math.pow((math.exp((x / n)) - t_0), 3.0), 0.3333333333333333)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-5)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-133)
		tmp = Float64(Float64(log1p(x) + Float64(Float64(0.5 * Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / n)) - log(x))) / n);
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = Float64(Float64(Float64(1.0 + Float64(log(x) / n)) / x) / n);
	else
		tmp = (Float64(exp(Float64(x / n)) - t_0) ^ 3.0) ^ 0.3333333333333333;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-5], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-133], N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], N[(N[(N[(1.0 + N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[Power[N[Power[N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 96.1%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. log-rec 99.9%

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

      2. mul-1-neg 99.9%

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

      3. neg-mul-1 99.9%

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

      4. mul-1-neg 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-rgt-identity 99.9%

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

      8. associate-/l* 99.9%

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

      9. exp-to-pow 99.9%

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

      10. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-133

   1. Initial program 38.7%

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

   3. Taylor expanded in n around inf 83.7%

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

      1. associate--l+ 83.7%

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

      2. log1p-define 83.7%

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

      3. +-commutative 83.7%

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

      4. associate--r+ 83.7%

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

      5. distribute-lft-out-- 83.7%

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

      6. div-sub 83.7%

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

      7. log1p-define 83.7%

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

   5. Simplified 83.7%

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

   if 2.0000000000000001e-133 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e-7

   1. Initial program 16.3%

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

   3. Taylor expanded in n around inf 44.9%

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

      1. associate--l+ 44.9%

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

      2. log1p-define 44.9%

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

      3. +-commutative 44.9%

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

      4. associate--r+ 44.9%

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

      5. distribute-lft-out-- 44.9%

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

      6. div-sub 44.9%

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

      7. log1p-define 44.9%

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

   5. Simplified 44.9%

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

   6. Taylor expanded in x around inf 68.2%

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

      1. +-commutative 68.2%

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

      2. mul-1-neg 68.2%

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

      3. log-rec 68.2%

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

      4. distribute-neg-frac 68.2%

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

      5. remove-double-neg 68.2%

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

   8. Simplified 68.2%

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

   if 1.9999999999999999e-7 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 47.1%

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

      1. add-cbrt-cube 47.1%

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

      2. pow3 47.1%

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

      3. pow-to-exp 47.1%

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

      4. un-div-inv 47.1%

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

      5. +-commutative 47.1%

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

      6. log1p-define 99.7%

         \[\leadsto \sqrt[3]{{\left(e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3}}} - {x}^{\left(\frac{1}{n}\right)} \]
   5. Step-by-step derivation

      1. rem-cbrt-cube 99.8%

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

      2. *-un-lft-identity 99.8%

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

      3. exp-prod 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. add-cbrt-cube 99.9%

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

      2. pow1/3 99.8%

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

      3. pow3 99.8%

         \[\leadsto {\color{blue}{\left({\left({\left(e^{1}\right)}^{\left(\frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. pow-exp 100.0%

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

      5. *-un-lft-identity 100.0%

         \[\leadsto {\left({\left(e^{\color{blue}{\frac{x}{n}}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 100.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1 + \frac{\log x}{n}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ t_2 := \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{n + \frac{x}{\frac{0.5 - t\_2}{n}}}{\frac{n \cdot x}{\frac{-0.5 + t\_2}{n}}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1 (- (pow (+ 1.0 x) (/ 1.0 n)) t_0))
        (t_2 (/ (+ -0.3333333333333333 (/ 0.25 x)) x)))
   (if (<= t_1 (- INFINITY))
     (- 1.0 t_0)
     (if (<= t_1 5e-10)
       (/ (log (/ (+ 1.0 x) x)) n)
       (/
        1.0
        (/
         x
         (/ (+ n (/ x (/ (- 0.5 t_2) n))) (/ (* n x) (/ (+ -0.5 t_2) n)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((1.0 + x), (1.0 / n)) - t_0;
	double t_2 = (-0.3333333333333333 + (0.25 / x)) / x;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 5e-10) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_2) / n))) / ((n * x) / ((-0.5 + t_2) / 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.pow((1.0 + x), (1.0 / n)) - t_0;
	double t_2 = (-0.3333333333333333 + (0.25 / x)) / x;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 5e-10) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_2) / n))) / ((n * x) / ((-0.5 + t_2) / n))));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((1.0 + x), (1.0 / n)) - t_0
	t_2 = (-0.3333333333333333 + (0.25 / x)) / x
	tmp = 0
	if t_1 <= -math.inf:
		tmp = 1.0 - t_0
	elif t_1 <= 5e-10:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_2) / n))) / ((n * x) / ((-0.5 + t_2) / n))))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - t_0)
	t_2 = Float64(Float64(-0.3333333333333333 + Float64(0.25 / x)) / x)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 5e-10)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(1.0 / Float64(x / Float64(Float64(n + Float64(x / Float64(Float64(0.5 - t_2) / n))) / Float64(Float64(n * x) / Float64(Float64(-0.5 + t_2) / n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((1.0 + x) ^ (1.0 / n)) - t_0;
	t_2 = (-0.3333333333333333 + (0.25 / x)) / x;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = 1.0 - t_0;
	elseif (t_1 <= 5e-10)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_2) / n))) / ((n * x) / ((-0.5 + t_2) / 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[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.3333333333333333 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 5e-10], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(1.0 / N[(x / N[(N[(n + N[(x / N[(N[(0.5 - t$95$2), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(n * x), $MachinePrecision] / N[(N[(-0.5 + t$95$2), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 5.00000000000000031e-10

   1. Initial program 47.0%

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

   3. Taylor expanded in n around inf 79.7%

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

      1. log1p-define 79.7%

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

   5. Simplified 79.7%

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

      1. log1p-undefine 79.7%

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

      2. diff-log 79.1%

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

   7. Applied egg-rr 79.1%

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

      1. +-commutative 79.1%

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

   9. Simplified 79.1%

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

   if 5.00000000000000031e-10 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 47.1%

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

   3. Taylor expanded in n around inf 5.9%

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

      1. log1p-define 5.9%

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

   5. Simplified 5.9%

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

   6. Taylor expanded in x around -inf 0.3%

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

      1. associate-*r/ 0.3%

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

   8. Simplified 0.3%

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

      1. clear-num 0.3%

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

      2. inv-pow 0.3%

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

   10. Applied egg-rr 50.7%

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

       1. unpow-1 50.7%

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

       2. sub-neg 50.7%

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

       3. *-lft-identity 50.7%

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

       4. *-lft-identity 50.7%

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

       5. associate-/l/ 50.7%

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

       6. sub-neg 50.7%

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

       7. metadata-eval 50.7%

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

       8. distribute-neg-frac 50.7%

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

       9. metadata-eval 50.7%

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

   12. Simplified 50.7%

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

       1. clear-num 50.7%

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

       2. frac-add 52.9%

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

       3. *-un-lft-identity 52.9%

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

       4. associate-/r* 52.9%

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

       5. sub-div 52.9%

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

       6. associate-/r* 52.9%

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

       7. sub-div 52.9%

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

   14. Applied egg-rr 52.9%

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

   16. Simplified 52.9%

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

4. Final simplification 77.5%

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

Alternative 4: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1 + \frac{\log x}{n}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(e^{\frac{x}{n}} - t\_0\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-5)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-133)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e-7)
         (/ (/ (+ 1.0 (/ (log x) n)) x) n)
         (pow (pow (- (exp (/ x n)) t_0) 3.0) 0.3333333333333333))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-133) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e-7) {
		tmp = ((1.0 + (log(x) / n)) / x) / n;
	} else {
		tmp = pow(pow((exp((x / n)) - t_0), 3.0), 0.3333333333333333);
	}
	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-5)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-133) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d-7) then
        tmp = ((1.0d0 + (log(x) / n)) / x) / n
    else
        tmp = ((exp((x / n)) - t_0) ** 3.0d0) ** 0.3333333333333333d0
    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-5) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-133) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e-7) {
		tmp = ((1.0 + (Math.log(x) / n)) / x) / n;
	} else {
		tmp = Math.pow(Math.pow((Math.exp((x / n)) - t_0), 3.0), 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-5:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-133:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e-7:
		tmp = ((1.0 + (math.log(x) / n)) / x) / n
	else:
		tmp = math.pow(math.pow((math.exp((x / n)) - t_0), 3.0), 0.3333333333333333)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-5)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-133)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = Float64(Float64(Float64(1.0 + Float64(log(x) / n)) / x) / n);
	else
		tmp = (Float64(exp(Float64(x / n)) - t_0) ^ 3.0) ^ 0.3333333333333333;
	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-5)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-133)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e-7)
		tmp = ((1.0 + (log(x) / n)) / x) / n;
	else
		tmp = ((exp((x / n)) - t_0) ^ 3.0) ^ 0.3333333333333333;
	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-5], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-133], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], N[(N[(N[(1.0 + N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[Power[N[Power[N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 96.1%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. log-rec 99.9%

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

      2. mul-1-neg 99.9%

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

      3. neg-mul-1 99.9%

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

      4. mul-1-neg 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-rgt-identity 99.9%

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

      8. associate-/l* 99.9%

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

      9. exp-to-pow 99.9%

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

      10. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-133

   1. Initial program 38.7%

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

   3. Taylor expanded in n around inf 83.2%

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

      1. log1p-define 83.2%

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

   5. Simplified 83.2%

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

      1. log1p-undefine 83.2%

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

      2. diff-log 83.2%

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

   7. Applied egg-rr 83.2%

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

      1. +-commutative 83.2%

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

   9. Simplified 83.2%

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

   if 2.0000000000000001e-133 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e-7

   1. Initial program 16.3%

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

   3. Taylor expanded in n around inf 44.9%

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

      1. associate--l+ 44.9%

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

      2. log1p-define 44.9%

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

      3. +-commutative 44.9%

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

      4. associate--r+ 44.9%

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

      5. distribute-lft-out-- 44.9%

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

      6. div-sub 44.9%

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

      7. log1p-define 44.9%

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

   5. Simplified 44.9%

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

   6. Taylor expanded in x around inf 68.2%

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

      1. +-commutative 68.2%

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

      2. mul-1-neg 68.2%

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

      3. log-rec 68.2%

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

      4. distribute-neg-frac 68.2%

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

      5. remove-double-neg 68.2%

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

   8. Simplified 68.2%

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

   if 1.9999999999999999e-7 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 47.1%

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

      1. add-cbrt-cube 47.1%

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

      2. pow3 47.1%

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

      3. pow-to-exp 47.1%

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

      4. un-div-inv 47.1%

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

      5. +-commutative 47.1%

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

      6. log1p-define 99.7%

         \[\leadsto \sqrt[3]{{\left(e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3}}} - {x}^{\left(\frac{1}{n}\right)} \]
   5. Step-by-step derivation

      1. rem-cbrt-cube 99.8%

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

      2. *-un-lft-identity 99.8%

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

      3. exp-prod 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. add-cbrt-cube 99.9%

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

      2. pow1/3 99.8%

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

      3. pow3 99.8%

         \[\leadsto {\color{blue}{\left({\left({\left(e^{1}\right)}^{\left(\frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. pow-exp 100.0%

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

      5. *-un-lft-identity 100.0%

         \[\leadsto {\left({\left(e^{\color{blue}{\frac{x}{n}}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 100.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1 + \frac{\log x}{n}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1 + \frac{\log x}{n}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\frac{x}{n}\right)} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-5)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-133)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e-7)
         (/ (/ (+ 1.0 (/ (log x) n)) x) n)
         (- (pow E (/ x n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-133) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e-7) {
		tmp = ((1.0 + (log(x) / n)) / x) / n;
	} else {
		tmp = pow(((double) M_E), (x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-133) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e-7) {
		tmp = ((1.0 + (Math.log(x) / n)) / x) / n;
	} else {
		tmp = Math.pow(Math.E, (x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-5:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-133:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e-7:
		tmp = ((1.0 + (math.log(x) / n)) / x) / n
	else:
		tmp = math.pow(math.e, (x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-5)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-133)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = Float64(Float64(Float64(1.0 + Float64(log(x) / n)) / x) / n);
	else
		tmp = Float64((exp(1) ^ Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-5)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-133)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e-7)
		tmp = ((1.0 + (log(x) / n)) / x) / n;
	else
		tmp = (2.71828182845904523536 ^ (x / n)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-5], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-133], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], N[(N[(N[(1.0 + N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[E, N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 96.1%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. log-rec 99.9%

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

      2. mul-1-neg 99.9%

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

      3. neg-mul-1 99.9%

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

      4. mul-1-neg 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-rgt-identity 99.9%

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

      8. associate-/l* 99.9%

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

      9. exp-to-pow 99.9%

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

      10. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-133

   1. Initial program 38.7%

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

   3. Taylor expanded in n around inf 83.2%

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

      1. log1p-define 83.2%

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

   5. Simplified 83.2%

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

      1. log1p-undefine 83.2%

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

      2. diff-log 83.2%

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

   7. Applied egg-rr 83.2%

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

      1. +-commutative 83.2%

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

   9. Simplified 83.2%

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

   if 2.0000000000000001e-133 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e-7

   1. Initial program 16.3%

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

   3. Taylor expanded in n around inf 44.9%

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

      1. associate--l+ 44.9%

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

      2. log1p-define 44.9%

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

      3. +-commutative 44.9%

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

      4. associate--r+ 44.9%

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

      5. distribute-lft-out-- 44.9%

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

      6. div-sub 44.9%

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

      7. log1p-define 44.9%

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

   5. Simplified 44.9%

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

   6. Taylor expanded in x around inf 68.2%

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

      1. +-commutative 68.2%

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

      2. mul-1-neg 68.2%

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

      3. log-rec 68.2%

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

      4. distribute-neg-frac 68.2%

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

      5. remove-double-neg 68.2%

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

   8. Simplified 68.2%

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

   if 1.9999999999999999e-7 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 47.1%

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

      1. add-cbrt-cube 47.1%

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

      2. pow3 47.1%

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

      3. pow-to-exp 47.1%

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

      4. un-div-inv 47.1%

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

      5. +-commutative 47.1%

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

      6. log1p-define 99.7%

         \[\leadsto \sqrt[3]{{\left(e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3}}} - {x}^{\left(\frac{1}{n}\right)} \]
   5. Step-by-step derivation

      1. rem-cbrt-cube 99.8%

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

      2. *-un-lft-identity 99.8%

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

      3. exp-prod 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-1-e 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. *-lft-identity 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1 + \frac{\log x}{n}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1 + \frac{\log x}{n}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-5)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-133)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e-7)
         (/ (/ (+ 1.0 (/ (log x) n)) x) n)
         (- (exp (/ x n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-133) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e-7) {
		tmp = ((1.0 + (log(x) / n)) / x) / n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1d-5)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-133) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d-7) then
        tmp = ((1.0d0 + (log(x) / n)) / x) / n
    else
        tmp = exp((x / n)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-133) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e-7) {
		tmp = ((1.0 + (Math.log(x) / n)) / x) / n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-5:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-133:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e-7:
		tmp = ((1.0 + (math.log(x) / n)) / x) / n
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-5)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-133)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = Float64(Float64(Float64(1.0 + Float64(log(x) / n)) / x) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-5)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-133)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e-7)
		tmp = ((1.0 + (log(x) / n)) / x) / n;
	else
		tmp = exp((x / n)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-5], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-133], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], N[(N[(N[(1.0 + N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 96.1%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. log-rec 99.9%

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

      2. mul-1-neg 99.9%

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

      3. neg-mul-1 99.9%

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

      4. mul-1-neg 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-rgt-identity 99.9%

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

      8. associate-/l* 99.9%

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

      9. exp-to-pow 99.9%

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

      10. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-133

   1. Initial program 38.7%

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

   3. Taylor expanded in n around inf 83.2%

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

      1. log1p-define 83.2%

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

   5. Simplified 83.2%

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

      1. log1p-undefine 83.2%

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

      2. diff-log 83.2%

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

   7. Applied egg-rr 83.2%

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

      1. +-commutative 83.2%

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

   9. Simplified 83.2%

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

   if 2.0000000000000001e-133 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e-7

   1. Initial program 16.3%

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

   3. Taylor expanded in n around inf 44.9%

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

      1. associate--l+ 44.9%

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

      2. log1p-define 44.9%

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

      3. +-commutative 44.9%

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

      4. associate--r+ 44.9%

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

      5. distribute-lft-out-- 44.9%

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

      6. div-sub 44.9%

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

      7. log1p-define 44.9%

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

   5. Simplified 44.9%

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

   6. Taylor expanded in x around inf 68.2%

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

      1. +-commutative 68.2%

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

      2. mul-1-neg 68.2%

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

      3. log-rec 68.2%

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

      4. distribute-neg-frac 68.2%

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

      5. remove-double-neg 68.2%

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

   8. Simplified 68.2%

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

   if 1.9999999999999999e-7 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 47.1%

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

      1. add-cbrt-cube 47.1%

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

      2. pow3 47.1%

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

      3. pow-to-exp 47.1%

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

      4. un-div-inv 47.1%

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

      5. +-commutative 47.1%

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

      6. log1p-define 99.7%

         \[\leadsto \sqrt[3]{{\left(e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3}}} - {x}^{\left(\frac{1}{n}\right)} \]
   5. Step-by-step derivation

      1. rem-cbrt-cube 99.8%

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

      2. *-un-lft-identity 99.8%

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

      3. exp-prod 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

   8. Taylor expanded in x around inf 99.8%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1 + \frac{\log x}{n}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{n}\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-5)
   (/ (pow x (/ 1.0 n)) (* n x))
   (if (<= (/ 1.0 n) 2e-133)
     (/ (log (/ (+ 1.0 x) x)) n)
     (/ (log1p (expm1 (/ 1.0 n))) x))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = pow(x, (1.0 / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-133) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = log1p(expm1((1.0 / n))) / x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = Math.pow(x, (1.0 / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-133) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = Math.log1p(Math.expm1((1.0 / n))) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e-5:
		tmp = math.pow(x, (1.0 / n)) / (n * x)
	elif (1.0 / n) <= 2e-133:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = math.log1p(math.expm1((1.0 / n))) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-5)
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-133)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(log1p(expm1(Float64(1.0 / n))) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-5], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-133], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Log[1 + N[(Exp[N[(1.0 / n), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 96.1%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. log-rec 99.9%

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

      2. mul-1-neg 99.9%

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

      3. neg-mul-1 99.9%

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

      4. mul-1-neg 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-rgt-identity 99.9%

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

      8. associate-/l* 99.9%

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

      9. exp-to-pow 99.9%

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

      10. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-133

   1. Initial program 38.7%

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

   3. Taylor expanded in n around inf 83.2%

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

      1. log1p-define 83.2%

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

   5. Simplified 83.2%

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

      1. log1p-undefine 83.2%

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

      2. diff-log 83.2%

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

   7. Applied egg-rr 83.2%

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

      1. +-commutative 83.2%

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

   9. Simplified 83.2%

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

   if 2.0000000000000001e-133 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.8%

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

   3. Taylor expanded in n around inf 19.8%

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

      1. log1p-define 19.8%

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

   5. Simplified 19.8%

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

   6. Taylor expanded in x around -inf 24.2%

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

      1. associate-*r/ 24.2%

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

   8. Simplified 24.2%

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

   9. Taylor expanded in x around inf 44.4%

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

       1. log1p-expm1-u 60.8%

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

   11. Applied egg-rr 60.8%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{n}\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{x}}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-5)
   (/ (pow x (/ 1.0 n)) (* n x))
   (if (<= (/ 1.0 n) 2e-133)
     (/ (log (/ (+ 1.0 x) x)) n)
     (log1p (expm1 (/ (/ 1.0 x) n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = pow(x, (1.0 / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-133) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = log1p(expm1(((1.0 / x) / n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = Math.pow(x, (1.0 / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-133) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = Math.log1p(Math.expm1(((1.0 / x) / n)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e-5:
		tmp = math.pow(x, (1.0 / n)) / (n * x)
	elif (1.0 / n) <= 2e-133:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = math.log1p(math.expm1(((1.0 / x) / n)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-5)
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-133)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = log1p(expm1(Float64(Float64(1.0 / x) / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-5], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-133], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[Log[1 + N[(Exp[N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 96.1%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. log-rec 99.9%

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

      2. mul-1-neg 99.9%

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

      3. neg-mul-1 99.9%

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

      4. mul-1-neg 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-rgt-identity 99.9%

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

      8. associate-/l* 99.9%

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

      9. exp-to-pow 99.9%

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

      10. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-133

   1. Initial program 38.7%

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

   3. Taylor expanded in n around inf 83.2%

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

      1. log1p-define 83.2%

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

   5. Simplified 83.2%

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

      1. log1p-undefine 83.2%

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

      2. diff-log 83.2%

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

   7. Applied egg-rr 83.2%

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

      1. +-commutative 83.2%

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

   9. Simplified 83.2%

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

   if 2.0000000000000001e-133 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.8%

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

   3. Taylor expanded in n around inf 19.8%

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

      1. log1p-define 19.8%

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

   5. Simplified 19.8%

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

   6. Taylor expanded in x around inf 43.6%

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

      1. *-commutative 43.6%

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

   8. Simplified 43.6%

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

      1. log1p-expm1-u 59.7%

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

      2. associate-/r* 60.5%

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

   10. Applied egg-rr 60.5%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{x}}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -200000000:\\ \;\;\;\;\frac{\frac{0.25}{n}}{{x}^{4}}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-264}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-180}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 8 \cdot 10^{+142}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{n + \frac{x}{\frac{0.5 - t\_0}{n}}}{\frac{n \cdot x}{\frac{-0.5 + t\_0}{n}}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (+ -0.3333333333333333 (/ 0.25 x)) x)))
   (if (<= (/ 1.0 n) -200000000.0)
     (/ (/ 0.25 n) (pow x 4.0))
     (if (<= (/ 1.0 n) 5e-264)
       (/ (/ 1.0 x) n)
       (if (<= (/ 1.0 n) 5e-180)
         (* (log x) (/ -1.0 n))
         (if (<= (/ 1.0 n) 2e-17)
           (/ (/ (+ 1.0 (/ (- (* 0.3333333333333333 (/ 1.0 x)) 0.5) x)) x) n)
           (if (<= (/ 1.0 n) 8e+142)
             (- 1.0 (pow x (/ 1.0 n)))
             (/
              1.0
              (/
               x
               (/
                (+ n (/ x (/ (- 0.5 t_0) n)))
                (/ (* n x) (/ (+ -0.5 t_0) n))))))))))))

C

double code(double x, double n) {
	double t_0 = (-0.3333333333333333 + (0.25 / x)) / x;
	double tmp;
	if ((1.0 / n) <= -200000000.0) {
		tmp = (0.25 / n) / pow(x, 4.0);
	} else if ((1.0 / n) <= 5e-264) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-180) {
		tmp = log(x) * (-1.0 / n);
	} else if ((1.0 / n) <= 2e-17) {
		tmp = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n;
	} else if ((1.0 / n) <= 8e+142) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_0) / n))) / ((n * x) / ((-0.5 + t_0) / n))));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-0.3333333333333333d0) + (0.25d0 / x)) / x
    if ((1.0d0 / n) <= (-200000000.0d0)) then
        tmp = (0.25d0 / n) / (x ** 4.0d0)
    else if ((1.0d0 / n) <= 5d-264) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 5d-180) then
        tmp = log(x) * ((-1.0d0) / n)
    else if ((1.0d0 / n) <= 2d-17) then
        tmp = ((1.0d0 + (((0.3333333333333333d0 * (1.0d0 / x)) - 0.5d0) / x)) / x) / n
    else if ((1.0d0 / n) <= 8d+142) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = 1.0d0 / (x / ((n + (x / ((0.5d0 - t_0) / n))) / ((n * x) / (((-0.5d0) + t_0) / n))))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (-0.3333333333333333 + (0.25 / x)) / x;
	double tmp;
	if ((1.0 / n) <= -200000000.0) {
		tmp = (0.25 / n) / Math.pow(x, 4.0);
	} else if ((1.0 / n) <= 5e-264) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-180) {
		tmp = Math.log(x) * (-1.0 / n);
	} else if ((1.0 / n) <= 2e-17) {
		tmp = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n;
	} else if ((1.0 / n) <= 8e+142) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_0) / n))) / ((n * x) / ((-0.5 + t_0) / n))));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (-0.3333333333333333 + (0.25 / x)) / x
	tmp = 0
	if (1.0 / n) <= -200000000.0:
		tmp = (0.25 / n) / math.pow(x, 4.0)
	elif (1.0 / n) <= 5e-264:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e-180:
		tmp = math.log(x) * (-1.0 / n)
	elif (1.0 / n) <= 2e-17:
		tmp = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n
	elif (1.0 / n) <= 8e+142:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_0) / n))) / ((n * x) / ((-0.5 + t_0) / n))))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-0.3333333333333333 + Float64(0.25 / x)) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -200000000.0)
		tmp = Float64(Float64(0.25 / n) / (x ^ 4.0));
	elseif (Float64(1.0 / n) <= 5e-264)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e-180)
		tmp = Float64(log(x) * Float64(-1.0 / n));
	elseif (Float64(1.0 / n) <= 2e-17)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(0.3333333333333333 * Float64(1.0 / x)) - 0.5) / x)) / x) / n);
	elseif (Float64(1.0 / n) <= 8e+142)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(1.0 / Float64(x / Float64(Float64(n + Float64(x / Float64(Float64(0.5 - t_0) / n))) / Float64(Float64(n * x) / Float64(Float64(-0.5 + t_0) / n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (-0.3333333333333333 + (0.25 / x)) / x;
	tmp = 0.0;
	if ((1.0 / n) <= -200000000.0)
		tmp = (0.25 / n) / (x ^ 4.0);
	elseif ((1.0 / n) <= 5e-264)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 5e-180)
		tmp = log(x) * (-1.0 / n);
	elseif ((1.0 / n) <= 2e-17)
		tmp = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n;
	elseif ((1.0 / n) <= 8e+142)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_0) / n))) / ((n * x) / ((-0.5 + t_0) / n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(-0.3333333333333333 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -200000000.0], N[(N[(0.25 / n), $MachinePrecision] / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-264], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-180], N[(N[Log[x], $MachinePrecision] * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-17], N[(N[(N[(1.0 + N[(N[(N[(0.3333333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 8e+142], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x / N[(N[(n + N[(x / N[(N[(0.5 - t$95$0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(n * x), $MachinePrecision] / N[(N[(-0.5 + t$95$0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e8

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 53.4%

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

      1. log1p-define 53.4%

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

   5. Simplified 53.4%

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

   6. Taylor expanded in x around -inf 1.8%

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

      1. associate-*r/ 1.8%

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

   8. Simplified 1.8%

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

      1. clear-num 1.8%

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

      2. inv-pow 1.8%

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

   10. Applied egg-rr 43.0%

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

       1. unpow-1 43.0%

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

       2. sub-neg 43.0%

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

       3. *-lft-identity 43.0%

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

       4. *-lft-identity 43.0%

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

       5. associate-/l/ 43.0%

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

       6. sub-neg 43.0%

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

       7. metadata-eval 43.0%

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

       8. distribute-neg-frac 43.0%

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

       9. metadata-eval 43.0%

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

   12. Simplified 43.0%

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

   13. Taylor expanded in x around 0 82.1%

       \[\leadsto \color{blue}{\frac{0.25}{n \cdot {x}^{4}}} \]
   14. Step-by-step derivation

       1. associate-/r* 82.1%

          \[\leadsto \color{blue}{\frac{\frac{0.25}{n}}{{x}^{4}}} \]

   15. Simplified 82.1%

       \[\leadsto \color{blue}{\frac{\frac{0.25}{n}}{{x}^{4}}} \]

   if -2e8 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e-264

   1. Initial program 37.6%

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

   3. Taylor expanded in n around inf 73.6%

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

      1. log1p-define 73.6%

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

   5. Simplified 73.6%

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

   6. Taylor expanded in x around inf 58.8%

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

   if 5.0000000000000001e-264 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e-180

   1. Initial program 35.0%

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

   3. Taylor expanded in n around inf 99.5%

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

      1. log1p-define 99.5%

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

   5. Simplified 99.5%

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

      1. div-inv 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in x around 0 70.9%

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

      1. neg-mul-1 70.9%

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

   10. Simplified 70.9%

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

   if 5.0000000000000001e-180 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000014e-17

   1. Initial program 25.9%

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

   3. Taylor expanded in n around inf 61.5%

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

      1. log1p-define 61.5%

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

   5. Simplified 61.5%

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

   6. Taylor expanded in x around -inf 65.6%

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

   if 2.00000000000000014e-17 < (/.f64 #s(literal 1 binary64) n) < 8.00000000000000041e142

   1. Initial program 75.2%

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

   3. Taylor expanded in x around 0 68.9%

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

      1. *-rgt-identity 68.9%

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

      2. associate-*l/ 68.9%

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

      3. associate-/l* 68.9%

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

      4. exp-to-pow 68.9%

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

   5. Simplified 68.9%

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

   if 8.00000000000000041e142 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.1%

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

   3. Taylor expanded in n around inf 5.9%

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

      1. log1p-define 5.9%

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

   5. Simplified 5.9%

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

   6. Taylor expanded in x around -inf 0.2%

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

      1. associate-*r/ 0.2%

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

   8. Simplified 0.2%

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

      1. clear-num 0.2%

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

      2. inv-pow 0.2%

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

   10. Applied egg-rr 69.0%

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

       1. unpow-1 69.0%

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

       2. sub-neg 69.0%

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

       3. *-lft-identity 69.0%

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

       4. *-lft-identity 69.0%

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

       5. associate-/l/ 69.0%

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

       6. sub-neg 69.0%

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

       7. metadata-eval 69.0%

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

       8. distribute-neg-frac 69.0%

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

       9. metadata-eval 69.0%

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

   12. Simplified 69.0%

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

       1. clear-num 69.0%

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

       2. frac-add 72.3%

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

       3. *-un-lft-identity 72.3%

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

       4. associate-/r* 72.3%

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

       5. sub-div 72.3%

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

       6. associate-/r* 72.3%

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

       7. sub-div 72.3%

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

   14. Applied egg-rr 72.3%

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

   16. Simplified 72.3%

       \[\leadsto \frac{1}{\frac{x}{\color{blue}{\frac{n + \frac{x}{\frac{0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{n}}}{\frac{x \cdot n}{\frac{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x} + -0.5}{n}}}}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -200000000:\\ \;\;\;\;\frac{\frac{0.25}{n}}{{x}^{4}}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-264}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-180}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 8 \cdot 10^{+142}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{n + \frac{x}{\frac{0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{n}}}{\frac{n \cdot x}{\frac{-0.5 + \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{n}}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1 + \frac{\log x}{n}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 8 \cdot 10^{+142}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{n + \frac{x}{\frac{0.5 - t\_1}{n}}}{\frac{n \cdot x}{\frac{-0.5 + t\_1}{n}}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (+ -0.3333333333333333 (/ 0.25 x)) x)))
   (if (<= (/ 1.0 n) -1e-5)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-133)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e-7)
         (/ (/ (+ 1.0 (/ (log x) n)) x) n)
         (if (<= (/ 1.0 n) 8e+142)
           (- (+ 1.0 (/ x n)) t_0)
           (/
            1.0
            (/
             x
             (/
              (+ n (/ x (/ (- 0.5 t_1) n)))
              (/ (* n x) (/ (+ -0.5 t_1) n)))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (-0.3333333333333333 + (0.25 / x)) / x;
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-133) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e-7) {
		tmp = ((1.0 + (log(x) / n)) / x) / n;
	} else if ((1.0 / n) <= 8e+142) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_1) / n))) / ((n * x) / ((-0.5 + t_1) / 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 = ((-0.3333333333333333d0) + (0.25d0 / x)) / x
    if ((1.0d0 / n) <= (-1d-5)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-133) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d-7) then
        tmp = ((1.0d0 + (log(x) / n)) / x) / n
    else if ((1.0d0 / n) <= 8d+142) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 1.0d0 / (x / ((n + (x / ((0.5d0 - t_1) / n))) / ((n * x) / (((-0.5d0) + t_1) / 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 = (-0.3333333333333333 + (0.25 / x)) / x;
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-133) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e-7) {
		tmp = ((1.0 + (Math.log(x) / n)) / x) / n;
	} else if ((1.0 / n) <= 8e+142) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_1) / n))) / ((n * x) / ((-0.5 + t_1) / n))));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (-0.3333333333333333 + (0.25 / x)) / x
	tmp = 0
	if (1.0 / n) <= -1e-5:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-133:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e-7:
		tmp = ((1.0 + (math.log(x) / n)) / x) / n
	elif (1.0 / n) <= 8e+142:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_1) / n))) / ((n * x) / ((-0.5 + t_1) / n))))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(-0.3333333333333333 + Float64(0.25 / x)) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-5)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-133)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = Float64(Float64(Float64(1.0 + Float64(log(x) / n)) / x) / n);
	elseif (Float64(1.0 / n) <= 8e+142)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(x / Float64(Float64(n + Float64(x / Float64(Float64(0.5 - t_1) / n))) / Float64(Float64(n * x) / Float64(Float64(-0.5 + t_1) / n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = (-0.3333333333333333 + (0.25 / x)) / x;
	tmp = 0.0;
	if ((1.0 / n) <= -1e-5)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-133)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e-7)
		tmp = ((1.0 + (log(x) / n)) / x) / n;
	elseif ((1.0 / n) <= 8e+142)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_1) / n))) / ((n * x) / ((-0.5 + t_1) / 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[(-0.3333333333333333 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-5], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-133], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], N[(N[(N[(1.0 + N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 8e+142], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(x / N[(N[(n + N[(x / N[(N[(0.5 - t$95$1), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(n * x), $MachinePrecision] / N[(N[(-0.5 + t$95$1), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 96.1%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. log-rec 99.9%

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

      2. mul-1-neg 99.9%

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

      3. neg-mul-1 99.9%

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

      4. mul-1-neg 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-rgt-identity 99.9%

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

      8. associate-/l* 99.9%

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

      9. exp-to-pow 99.9%

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

      10. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-133

   1. Initial program 38.7%

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

   3. Taylor expanded in n around inf 83.2%

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

      1. log1p-define 83.2%

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

   5. Simplified 83.2%

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

      1. log1p-undefine 83.2%

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

      2. diff-log 83.2%

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

   7. Applied egg-rr 83.2%

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

      1. +-commutative 83.2%

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

   9. Simplified 83.2%

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

   if 2.0000000000000001e-133 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e-7

   1. Initial program 16.3%

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

   3. Taylor expanded in n around inf 44.9%

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

      1. associate--l+ 44.9%

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

      2. log1p-define 44.9%

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

      3. +-commutative 44.9%

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

      4. associate--r+ 44.9%

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

      5. distribute-lft-out-- 44.9%

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

      6. div-sub 44.9%

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

      7. log1p-define 44.9%

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

   5. Simplified 44.9%

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

   6. Taylor expanded in x around inf 68.2%

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

      1. +-commutative 68.2%

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

      2. mul-1-neg 68.2%

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

      3. log-rec 68.2%

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

      4. distribute-neg-frac 68.2%

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

      5. remove-double-neg 68.2%

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

   8. Simplified 68.2%

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

   if 1.9999999999999999e-7 < (/.f64 #s(literal 1 binary64) n) < 8.00000000000000041e142

   1. Initial program 80.6%

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

   3. Taylor expanded in x around 0 74.3%

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

   if 8.00000000000000041e142 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.1%

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

   3. Taylor expanded in n around inf 5.9%

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

      1. log1p-define 5.9%

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

   5. Simplified 5.9%

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

   6. Taylor expanded in x around -inf 0.2%

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

      1. associate-*r/ 0.2%

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

   8. Simplified 0.2%

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

      1. clear-num 0.2%

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

      2. inv-pow 0.2%

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

   10. Applied egg-rr 69.0%

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

       1. unpow-1 69.0%

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

       2. sub-neg 69.0%

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

       3. *-lft-identity 69.0%

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

       4. *-lft-identity 69.0%

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

       5. associate-/l/ 69.0%

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

       6. sub-neg 69.0%

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

       7. metadata-eval 69.0%

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

       8. distribute-neg-frac 69.0%

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

       9. metadata-eval 69.0%

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

   12. Simplified 69.0%

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

       1. clear-num 69.0%

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

       2. frac-add 72.3%

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

       3. *-un-lft-identity 72.3%

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

       4. associate-/r* 72.3%

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

       5. sub-div 72.3%

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

       6. associate-/r* 72.3%

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

       7. sub-div 72.3%

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

   14. Applied egg-rr 72.3%

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

   16. Simplified 72.3%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1 + \frac{\log x}{n}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 8 \cdot 10^{+142}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{n + \frac{x}{\frac{0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{n}}}{\frac{n \cdot x}{\frac{-0.5 + \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{n}}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1 + \frac{\log x}{n}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 8 \cdot 10^{+142}:\\ \;\;\;\;1 - t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{n + \frac{x}{\frac{0.5 - t\_0}{n}}}{\frac{n \cdot x}{\frac{-0.5 + t\_0}{n}}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (+ -0.3333333333333333 (/ 0.25 x)) x)) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-5)
     (/ t_1 (* n x))
     (if (<= (/ 1.0 n) 2e-133)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e-7)
         (/ (/ (+ 1.0 (/ (log x) n)) x) n)
         (if (<= (/ 1.0 n) 8e+142)
           (- 1.0 t_1)
           (/
            1.0
            (/
             x
             (/
              (+ n (/ x (/ (- 0.5 t_0) n)))
              (/ (* n x) (/ (+ -0.5 t_0) n)))))))))))

C

double code(double x, double n) {
	double t_0 = (-0.3333333333333333 + (0.25 / x)) / x;
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = t_1 / (n * x);
	} else if ((1.0 / n) <= 2e-133) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e-7) {
		tmp = ((1.0 + (log(x) / n)) / x) / n;
	} else if ((1.0 / n) <= 8e+142) {
		tmp = 1.0 - t_1;
	} else {
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_0) / n))) / ((n * x) / ((-0.5 + t_0) / 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 = ((-0.3333333333333333d0) + (0.25d0 / x)) / x
    t_1 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1d-5)) then
        tmp = t_1 / (n * x)
    else if ((1.0d0 / n) <= 2d-133) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d-7) then
        tmp = ((1.0d0 + (log(x) / n)) / x) / n
    else if ((1.0d0 / n) <= 8d+142) then
        tmp = 1.0d0 - t_1
    else
        tmp = 1.0d0 / (x / ((n + (x / ((0.5d0 - t_0) / n))) / ((n * x) / (((-0.5d0) + t_0) / n))))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (-0.3333333333333333 + (0.25 / x)) / x;
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = t_1 / (n * x);
	} else if ((1.0 / n) <= 2e-133) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e-7) {
		tmp = ((1.0 + (Math.log(x) / n)) / x) / n;
	} else if ((1.0 / n) <= 8e+142) {
		tmp = 1.0 - t_1;
	} else {
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_0) / n))) / ((n * x) / ((-0.5 + t_0) / n))));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (-0.3333333333333333 + (0.25 / x)) / x
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-5:
		tmp = t_1 / (n * x)
	elif (1.0 / n) <= 2e-133:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e-7:
		tmp = ((1.0 + (math.log(x) / n)) / x) / n
	elif (1.0 / n) <= 8e+142:
		tmp = 1.0 - t_1
	else:
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_0) / n))) / ((n * x) / ((-0.5 + t_0) / n))))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-0.3333333333333333 + Float64(0.25 / x)) / x)
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-5)
		tmp = Float64(t_1 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-133)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = Float64(Float64(Float64(1.0 + Float64(log(x) / n)) / x) / n);
	elseif (Float64(1.0 / n) <= 8e+142)
		tmp = Float64(1.0 - t_1);
	else
		tmp = Float64(1.0 / Float64(x / Float64(Float64(n + Float64(x / Float64(Float64(0.5 - t_0) / n))) / Float64(Float64(n * x) / Float64(Float64(-0.5 + t_0) / n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (-0.3333333333333333 + (0.25 / x)) / x;
	t_1 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-5)
		tmp = t_1 / (n * x);
	elseif ((1.0 / n) <= 2e-133)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e-7)
		tmp = ((1.0 + (log(x) / n)) / x) / n;
	elseif ((1.0 / n) <= 8e+142)
		tmp = 1.0 - t_1;
	else
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_0) / n))) / ((n * x) / ((-0.5 + t_0) / n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(-0.3333333333333333 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-5], N[(t$95$1 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-133], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], N[(N[(N[(1.0 + N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 8e+142], N[(1.0 - t$95$1), $MachinePrecision], N[(1.0 / N[(x / N[(N[(n + N[(x / N[(N[(0.5 - t$95$0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(n * x), $MachinePrecision] / N[(N[(-0.5 + t$95$0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 96.1%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. log-rec 99.9%

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

      2. mul-1-neg 99.9%

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

      3. neg-mul-1 99.9%

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

      4. mul-1-neg 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-rgt-identity 99.9%

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

      8. associate-/l* 99.9%

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

      9. exp-to-pow 99.9%

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

      10. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-133

   1. Initial program 38.7%

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

   3. Taylor expanded in n around inf 83.2%

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

      1. log1p-define 83.2%

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

   5. Simplified 83.2%

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

      1. log1p-undefine 83.2%

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

      2. diff-log 83.2%

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

   7. Applied egg-rr 83.2%

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

      1. +-commutative 83.2%

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

   9. Simplified 83.2%

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

   if 2.0000000000000001e-133 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e-7

   1. Initial program 16.3%

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

   3. Taylor expanded in n around inf 44.9%

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

      1. associate--l+ 44.9%

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

      2. log1p-define 44.9%

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

      3. +-commutative 44.9%

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

      4. associate--r+ 44.9%

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

      5. distribute-lft-out-- 44.9%

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

      6. div-sub 44.9%

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

      7. log1p-define 44.9%

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

   5. Simplified 44.9%

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

   6. Taylor expanded in x around inf 68.2%

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

      1. +-commutative 68.2%

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

      2. mul-1-neg 68.2%

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

      3. log-rec 68.2%

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

      4. distribute-neg-frac 68.2%

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

      5. remove-double-neg 68.2%

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

   8. Simplified 68.2%

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

   if 1.9999999999999999e-7 < (/.f64 #s(literal 1 binary64) n) < 8.00000000000000041e142

   1. Initial program 80.6%

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

   3. Taylor expanded in x around 0 74.2%

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

      1. *-rgt-identity 74.2%

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

      2. associate-*l/ 74.2%

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

      3. associate-/l* 74.2%

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

      4. exp-to-pow 74.2%

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

   5. Simplified 74.2%

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

   if 8.00000000000000041e142 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.1%

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

   3. Taylor expanded in n around inf 5.9%

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

      1. log1p-define 5.9%

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

   5. Simplified 5.9%

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

   6. Taylor expanded in x around -inf 0.2%

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

      1. associate-*r/ 0.2%

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

   8. Simplified 0.2%

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

      1. clear-num 0.2%

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

      2. inv-pow 0.2%

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

   10. Applied egg-rr 69.0%

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

       1. unpow-1 69.0%

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

       2. sub-neg 69.0%

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

       3. *-lft-identity 69.0%

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

       4. *-lft-identity 69.0%

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

       5. associate-/l/ 69.0%

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

       6. sub-neg 69.0%

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

       7. metadata-eval 69.0%

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

       8. distribute-neg-frac 69.0%

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

       9. metadata-eval 69.0%

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

   12. Simplified 69.0%

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

       1. clear-num 69.0%

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

       2. frac-add 72.3%

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

       3. *-un-lft-identity 72.3%

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

       4. associate-/r* 72.3%

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

       5. sub-div 72.3%

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

       6. associate-/r* 72.3%

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

       7. sub-div 72.3%

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

   14. Applied egg-rr 72.3%

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

   16. Simplified 72.3%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1 + \frac{\log x}{n}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 8 \cdot 10^{+142}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{n + \frac{x}{\frac{0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{n}}}{\frac{n \cdot x}{\frac{-0.5 + \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{n}}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 8 \cdot 10^{+142}:\\ \;\;\;\;1 - t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{n + \frac{x}{\frac{0.5 - t\_0}{n}}}{\frac{n \cdot x}{\frac{-0.5 + t\_0}{n}}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (+ -0.3333333333333333 (/ 0.25 x)) x)) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-5)
     (/ t_1 (* n x))
     (if (<= (/ 1.0 n) 2e-133)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e-17)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 8e+142)
           (- 1.0 t_1)
           (/
            1.0
            (/
             x
             (/
              (+ n (/ x (/ (- 0.5 t_0) n)))
              (/ (* n x) (/ (+ -0.5 t_0) n)))))))))))

C

double code(double x, double n) {
	double t_0 = (-0.3333333333333333 + (0.25 / x)) / x;
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = t_1 / (n * x);
	} else if ((1.0 / n) <= 2e-133) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 8e+142) {
		tmp = 1.0 - t_1;
	} else {
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_0) / n))) / ((n * x) / ((-0.5 + t_0) / 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 = ((-0.3333333333333333d0) + (0.25d0 / x)) / x
    t_1 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1d-5)) then
        tmp = t_1 / (n * x)
    else if ((1.0d0 / n) <= 2d-133) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d-17) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 8d+142) then
        tmp = 1.0d0 - t_1
    else
        tmp = 1.0d0 / (x / ((n + (x / ((0.5d0 - t_0) / n))) / ((n * x) / (((-0.5d0) + t_0) / n))))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (-0.3333333333333333 + (0.25 / x)) / x;
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = t_1 / (n * x);
	} else if ((1.0 / n) <= 2e-133) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 8e+142) {
		tmp = 1.0 - t_1;
	} else {
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_0) / n))) / ((n * x) / ((-0.5 + t_0) / n))));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (-0.3333333333333333 + (0.25 / x)) / x
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-5:
		tmp = t_1 / (n * x)
	elif (1.0 / n) <= 2e-133:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e-17:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 8e+142:
		tmp = 1.0 - t_1
	else:
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_0) / n))) / ((n * x) / ((-0.5 + t_0) / n))))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-0.3333333333333333 + Float64(0.25 / x)) / x)
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-5)
		tmp = Float64(t_1 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-133)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e-17)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 8e+142)
		tmp = Float64(1.0 - t_1);
	else
		tmp = Float64(1.0 / Float64(x / Float64(Float64(n + Float64(x / Float64(Float64(0.5 - t_0) / n))) / Float64(Float64(n * x) / Float64(Float64(-0.5 + t_0) / n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (-0.3333333333333333 + (0.25 / x)) / x;
	t_1 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-5)
		tmp = t_1 / (n * x);
	elseif ((1.0 / n) <= 2e-133)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e-17)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 8e+142)
		tmp = 1.0 - t_1;
	else
		tmp = 1.0 / (x / ((n + (x / ((0.5 - t_0) / n))) / ((n * x) / ((-0.5 + t_0) / n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(-0.3333333333333333 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-5], N[(t$95$1 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-133], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-17], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 8e+142], N[(1.0 - t$95$1), $MachinePrecision], N[(1.0 / N[(x / N[(N[(n + N[(x / N[(N[(0.5 - t$95$0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(n * x), $MachinePrecision] / N[(N[(-0.5 + t$95$0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 96.1%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. log-rec 99.9%

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

      2. mul-1-neg 99.9%

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

      3. neg-mul-1 99.9%

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

      4. mul-1-neg 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-rgt-identity 99.9%

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

      8. associate-/l* 99.9%

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

      9. exp-to-pow 99.9%

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

      10. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-133

   1. Initial program 38.7%

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

   3. Taylor expanded in n around inf 83.2%

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

      1. log1p-define 83.2%

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

   5. Simplified 83.2%

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

      1. log1p-undefine 83.2%

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

      2. diff-log 83.2%

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

   7. Applied egg-rr 83.2%

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

      1. +-commutative 83.2%

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

   9. Simplified 83.2%

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

   if 2.0000000000000001e-133 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000014e-17

   1. Initial program 14.7%

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

   3. Taylor expanded in n around inf 43.9%

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

      1. log1p-define 43.9%

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

   5. Simplified 43.9%

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

   6. Taylor expanded in x around inf 70.8%

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

   if 2.00000000000000014e-17 < (/.f64 #s(literal 1 binary64) n) < 8.00000000000000041e142

   1. Initial program 75.2%

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

   3. Taylor expanded in x around 0 68.9%

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

      1. *-rgt-identity 68.9%

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

      2. associate-*l/ 68.9%

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

      3. associate-/l* 68.9%

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

      4. exp-to-pow 68.9%

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

   5. Simplified 68.9%

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

   if 8.00000000000000041e142 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.1%

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

   3. Taylor expanded in n around inf 5.9%

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

      1. log1p-define 5.9%

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

   5. Simplified 5.9%

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

   6. Taylor expanded in x around -inf 0.2%

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

      1. associate-*r/ 0.2%

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

   8. Simplified 0.2%

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

      1. clear-num 0.2%

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

      2. inv-pow 0.2%

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

   10. Applied egg-rr 69.0%

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

       1. unpow-1 69.0%

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

       2. sub-neg 69.0%

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

       3. *-lft-identity 69.0%

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

       4. *-lft-identity 69.0%

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

       5. associate-/l/ 69.0%

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

       6. sub-neg 69.0%

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

       7. metadata-eval 69.0%

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

       8. distribute-neg-frac 69.0%

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

       9. metadata-eval 69.0%

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

   12. Simplified 69.0%

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

       1. clear-num 69.0%

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

       2. frac-add 72.3%

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

       3. *-un-lft-identity 72.3%

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

       4. associate-/r* 72.3%

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

       5. sub-div 72.3%

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

       6. associate-/r* 72.3%

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

       7. sub-div 72.3%

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

   14. Applied egg-rr 72.3%

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

   16. Simplified 72.3%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 8 \cdot 10^{+142}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{n + \frac{x}{\frac{0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{n}}}{\frac{n \cdot x}{\frac{-0.5 + \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{n}}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+119}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{\frac{0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{n}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{n \cdot {x}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.5e-88)
   (/ (log x) (- n))
   (if (<= x 7.8e+119)
     (*
      (/ 1.0 x)
      (-
       (/ 1.0 n)
       (/ (/ (- 0.5 (/ (+ -0.3333333333333333 (/ 0.25 x)) x)) n) x)))
     (/ -0.25 (* n (pow x 4.0))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.5e-88) {
		tmp = log(x) / -n;
	} else if (x <= 7.8e+119) {
		tmp = (1.0 / x) * ((1.0 / n) - (((0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / n) / x));
	} else {
		tmp = -0.25 / (n * pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 3.5d-88) then
        tmp = log(x) / -n
    else if (x <= 7.8d+119) then
        tmp = (1.0d0 / x) * ((1.0d0 / n) - (((0.5d0 - (((-0.3333333333333333d0) + (0.25d0 / x)) / x)) / n) / x))
    else
        tmp = (-0.25d0) / (n * (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.5e-88) {
		tmp = Math.log(x) / -n;
	} else if (x <= 7.8e+119) {
		tmp = (1.0 / x) * ((1.0 / n) - (((0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / n) / x));
	} else {
		tmp = -0.25 / (n * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.5e-88:
		tmp = math.log(x) / -n
	elif x <= 7.8e+119:
		tmp = (1.0 / x) * ((1.0 / n) - (((0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / n) / x))
	else:
		tmp = -0.25 / (n * math.pow(x, 4.0))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.5e-88)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 7.8e+119)
		tmp = Float64(Float64(1.0 / x) * Float64(Float64(1.0 / n) - Float64(Float64(Float64(0.5 - Float64(Float64(-0.3333333333333333 + Float64(0.25 / x)) / x)) / n) / x)));
	else
		tmp = Float64(-0.25 / Float64(n * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 3.5e-88)
		tmp = log(x) / -n;
	elseif (x <= 7.8e+119)
		tmp = (1.0 / x) * ((1.0 / n) - (((0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / n) / x));
	else
		tmp = -0.25 / (n * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.5e-88], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 7.8e+119], N[(N[(1.0 / x), $MachinePrecision] * N[(N[(1.0 / n), $MachinePrecision] - N[(N[(N[(0.5 - N[(N[(-0.3333333333333333 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 / N[(n * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.5000000000000001e-88

   1. Initial program 42.2%

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

   3. Taylor expanded in n around inf 51.5%

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

      1. log1p-define 51.5%

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

   5. Simplified 51.5%

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

   6. Taylor expanded in x around 0 51.5%

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

      1. neg-mul-1 51.5%

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

   8. Simplified 51.5%

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

   if 3.5000000000000001e-88 < x < 7.7999999999999997e119

   1. Initial program 43.8%

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

   3. Taylor expanded in n around inf 42.8%

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

      1. log1p-define 42.8%

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

   5. Simplified 42.8%

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

   6. Taylor expanded in x around -inf 35.7%

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

      1. associate-*r/ 35.7%

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

   8. Simplified 35.7%

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

      1. clear-num 34.7%

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

      2. inv-pow 34.7%

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

   10. Applied egg-rr 29.9%

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

       1. unpow-1 29.9%

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

       2. sub-neg 29.9%

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

       3. *-lft-identity 29.9%

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

       4. *-lft-identity 29.9%

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

       5. associate-/l/ 29.9%

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

       6. sub-neg 29.9%

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

       7. metadata-eval 29.9%

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

       8. distribute-neg-frac 29.9%

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

       9. metadata-eval 29.9%

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

   12. Simplified 29.9%

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

       1. associate-/r/ 29.8%

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

       2. +-commutative 29.8%

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

       3. add-sqr-sqrt 12.4%

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

       4. sqrt-unprod 36.4%

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

       5. frac-times 36.3%

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

       6. metadata-eval 36.3%

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

       7. metadata-eval 36.3%

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

       8. frac-times 36.4%

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

       9. sqrt-unprod 31.9%

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

       10. add-sqr-sqrt 56.3%

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

       11. associate-/r* 56.3%

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

       12. sub-div 56.3%

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

   14. Applied egg-rr 56.3%

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

   if 7.7999999999999997e119 < x

   1. Initial program 81.0%

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

   3. Taylor expanded in n around inf 81.0%

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

      1. log1p-define 81.0%

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

   5. Simplified 81.0%

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

   6. Taylor expanded in x around -inf 67.1%

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

      1. associate-*r/ 67.1%

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

   8. Simplified 67.1%

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

   9. Taylor expanded in x around 0 81.0%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+119}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{\frac{0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{n}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{n \cdot {x}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.95 \cdot 10^{-88}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{\frac{0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{n}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.95e-88)
   (/ (log x) (- n))
   (*
    (/ 1.0 x)
    (-
     (/ 1.0 n)
     (/ (/ (- 0.5 (/ (+ -0.3333333333333333 (/ 0.25 x)) x)) n) x)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.95e-88) {
		tmp = log(x) / -n;
	} else {
		tmp = (1.0 / x) * ((1.0 / n) - (((0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / n) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 3.95d-88) then
        tmp = log(x) / -n
    else
        tmp = (1.0d0 / x) * ((1.0d0 / n) - (((0.5d0 - (((-0.3333333333333333d0) + (0.25d0 / x)) / x)) / n) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.95e-88) {
		tmp = Math.log(x) / -n;
	} else {
		tmp = (1.0 / x) * ((1.0 / n) - (((0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / n) / x));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.95e-88:
		tmp = math.log(x) / -n
	else:
		tmp = (1.0 / x) * ((1.0 / n) - (((0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / n) / x))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.95e-88)
		tmp = Float64(log(x) / Float64(-n));
	else
		tmp = Float64(Float64(1.0 / x) * Float64(Float64(1.0 / n) - Float64(Float64(Float64(0.5 - Float64(Float64(-0.3333333333333333 + Float64(0.25 / x)) / x)) / n) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 3.95e-88)
		tmp = log(x) / -n;
	else
		tmp = (1.0 / x) * ((1.0 / n) - (((0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / n) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.95e-88], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] * N[(N[(1.0 / n), $MachinePrecision] - N[(N[(N[(0.5 - N[(N[(-0.3333333333333333 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.94999999999999983e-88

   1. Initial program 42.2%

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

   3. Taylor expanded in n around inf 51.5%

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

      1. log1p-define 51.5%

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

   5. Simplified 51.5%

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

   6. Taylor expanded in x around 0 51.5%

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

      1. neg-mul-1 51.5%

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

   8. Simplified 51.5%

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

   if 3.94999999999999983e-88 < x

   1. Initial program 61.8%

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

   3. Taylor expanded in n around inf 61.3%

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

      1. log1p-define 61.3%

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

   5. Simplified 61.3%

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

   6. Taylor expanded in x around -inf 50.9%

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

      1. associate-*r/ 50.9%

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

   8. Simplified 50.9%

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

      1. clear-num 48.9%

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

      2. inv-pow 48.9%

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

   10. Applied egg-rr 40.2%

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

       1. unpow-1 40.2%

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

       2. sub-neg 40.2%

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

       3. *-lft-identity 40.2%

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

       4. *-lft-identity 40.2%

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

       5. associate-/l/ 40.2%

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

       6. sub-neg 40.2%

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

       7. metadata-eval 40.2%

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

       8. distribute-neg-frac 40.2%

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

       9. metadata-eval 40.2%

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

   12. Simplified 40.2%

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

       1. associate-/r/ 40.1%

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

       2. +-commutative 40.1%

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

       3. add-sqr-sqrt 18.5%

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

       4. sqrt-unprod 45.9%

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

       5. frac-times 45.9%

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

       6. metadata-eval 45.9%

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

       7. metadata-eval 45.9%

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

       8. frac-times 45.9%

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

       9. sqrt-unprod 31.5%

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

       10. add-sqr-sqrt 61.5%

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

       11. associate-/r* 61.5%

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

       12. sub-div 61.5%

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

   14. Applied egg-rr 61.5%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.95 \cdot 10^{-88}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{\frac{0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{n}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.4% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \cdot \left(\frac{1}{n} - \frac{\frac{0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{n}}{x}\right) \end{array} \]

FPCore

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

C

double code(double x, double n) {
	return (1.0 / x) * ((1.0 / n) - (((0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / n) / x));
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 / x) * ((1.0d0 / n) - (((0.5d0 - (((-0.3333333333333333d0) + (0.25d0 / x)) / x)) / n) / x))
end function

Java

public static double code(double x, double n) {
	return (1.0 / x) * ((1.0 / n) - (((0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / n) / x));
}

Python

def code(x, n):
	return (1.0 / x) * ((1.0 / n) - (((0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / n) / x))

Julia

function code(x, n)
	return Float64(Float64(1.0 / x) * Float64(Float64(1.0 / n) - Float64(Float64(Float64(0.5 - Float64(Float64(-0.3333333333333333 + Float64(0.25 / x)) / x)) / n) / x)))
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / x) * ((1.0 / n) - (((0.5 - ((-0.3333333333333333 + (0.25 / x)) / x)) / n) / x));
end

Wolfram

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

Derivation

1. Initial program 54.0%

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

3. Taylor expanded in n around inf 57.4%

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

   1. log1p-define 57.4%

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

5. Simplified 57.4%

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

6. Taylor expanded in x around -inf 31.1%

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

   1. associate-*r/ 31.1%

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

8. Simplified 31.1%

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

   1. clear-num 29.9%

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

   2. inv-pow 29.9%

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

10. Applied egg-rr 37.9%

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

    1. unpow-1 37.9%

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

    2. sub-neg 37.9%

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

    3. *-lft-identity 37.9%

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

    4. *-lft-identity 37.9%

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

    5. associate-/l/ 37.9%

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

    6. sub-neg 37.9%

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

    7. metadata-eval 37.9%

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

    8. distribute-neg-frac 37.9%

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

    9. metadata-eval 37.9%

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

12. Simplified 37.9%

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

    1. associate-/r/ 37.9%

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

    2. +-commutative 37.9%

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

    3. add-sqr-sqrt 20.0%

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

    4. sqrt-unprod 37.0%

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

    5. frac-times 37.0%

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

    6. metadata-eval 37.0%

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

    7. metadata-eval 37.0%

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

    8. frac-times 37.0%

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

    9. sqrt-unprod 23.8%

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

    10. add-sqr-sqrt 50.8%

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

    11. associate-/r* 50.8%

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

    12. sub-div 50.8%

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

14. Applied egg-rr 50.8%

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

15. Final simplification 50.8%

    \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{n} - \frac{\frac{0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{n}}{x}\right) \]
16. Add Preprocessing

Alternative 16: 47.4% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n} + \frac{-0.5 + \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{n \cdot x}}{x} \end{array} \]

FPCore

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

C

double code(double x, double n) {
	return ((1.0 / n) + ((-0.5 + ((-0.3333333333333333 + (0.25 / x)) / x)) / (n * x))) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 / n) + (((-0.5d0) + (((-0.3333333333333333d0) + (0.25d0 / x)) / x)) / (n * x))) / x
end function

Java

public static double code(double x, double n) {
	return ((1.0 / n) + ((-0.5 + ((-0.3333333333333333 + (0.25 / x)) / x)) / (n * x))) / x;
}

Python

def code(x, n):
	return ((1.0 / n) + ((-0.5 + ((-0.3333333333333333 + (0.25 / x)) / x)) / (n * x))) / x

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 / n) + Float64(Float64(-0.5 + Float64(Float64(-0.3333333333333333 + Float64(0.25 / x)) / x)) / Float64(n * x))) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 / n) + ((-0.5 + ((-0.3333333333333333 + (0.25 / x)) / x)) / (n * x))) / x;
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(-0.5 + N[(N[(-0.3333333333333333 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 54.0%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 57.4%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-define 57.4%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

5. Simplified 57.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

6. Taylor expanded in x around -inf 31.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
7. Step-by-step derivation

   1. associate-*r/ 31.1%

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}\right)}{x}} \]

8. Simplified 31.1%

   \[\leadsto \color{blue}{\frac{-\left(\frac{-\left(\frac{-\left(\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}\right)}{x} - \frac{0.5}{n}\right)}{x} - \frac{1}{n}\right)}{x}} \]
9. Step-by-step derivation

   1. clear-num 29.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{-\left(\frac{-\left(\frac{-\left(\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}\right)}{x} - \frac{0.5}{n}\right)}{x} - \frac{1}{n}\right)}}} \]

   2. inv-pow 29.9%

      \[\leadsto \color{blue}{{\left(\frac{x}{-\left(\frac{-\left(\frac{-\left(\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}\right)}{x} - \frac{0.5}{n}\right)}{x} - \frac{1}{n}\right)}\right)}^{-1}} \]

10. Applied egg-rr 37.9%

    \[\leadsto \color{blue}{{\left(\frac{x}{\frac{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x} - \frac{1}{n}}\right)}^{-1}} \]
11. Step-by-step derivation

    1. unpow-1 37.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x} - \frac{1}{n}}}} \]

    2. sub-neg 37.9%

       \[\leadsto \frac{1}{\frac{x}{\color{blue}{\frac{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x} + \left(-\frac{1}{n}\right)}}} \]

    3. *-lft-identity 37.9%

       \[\leadsto \frac{1}{\frac{x}{\color{blue}{1 \cdot \frac{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x}} + \left(-\frac{1}{n}\right)}} \]

    4. *-lft-identity 37.9%

       \[\leadsto \frac{1}{\frac{x}{\color{blue}{\frac{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x}} + \left(-\frac{1}{n}\right)}} \]

    5. associate-/l/ 37.9%

       \[\leadsto \frac{1}{\frac{x}{\frac{\color{blue}{\frac{\frac{0.25}{x} - 0.3333333333333333}{x \cdot n}} - \frac{0.5}{n}}{x} + \left(-\frac{1}{n}\right)}} \]

    6. sub-neg 37.9%

       \[\leadsto \frac{1}{\frac{x}{\frac{\frac{\color{blue}{\frac{0.25}{x} + \left(-0.3333333333333333\right)}}{x \cdot n} - \frac{0.5}{n}}{x} + \left(-\frac{1}{n}\right)}} \]

    7. metadata-eval 37.9%

       \[\leadsto \frac{1}{\frac{x}{\frac{\frac{\frac{0.25}{x} + \color{blue}{-0.3333333333333333}}{x \cdot n} - \frac{0.5}{n}}{x} + \left(-\frac{1}{n}\right)}} \]

    8. distribute-neg-frac 37.9%

       \[\leadsto \frac{1}{\frac{x}{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x} + \color{blue}{\frac{-1}{n}}}} \]

    9. metadata-eval 37.9%

       \[\leadsto \frac{1}{\frac{x}{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x} + \frac{\color{blue}{-1}}{n}}} \]

12. Simplified 37.9%

    \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x} + \frac{-1}{n}}}} \]
13. Step-by-step derivation

    1. *-un-lft-identity 37.9%

       \[\leadsto \color{blue}{1 \cdot \frac{1}{\frac{x}{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x} + \frac{-1}{n}}}} \]

    2. associate-/r/ 37.9%

       \[\leadsto 1 \cdot \color{blue}{\left(\frac{1}{x} \cdot \left(\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x} + \frac{-1}{n}\right)\right)} \]

    3. +-commutative 37.9%

       \[\leadsto 1 \cdot \left(\frac{1}{x} \cdot \color{blue}{\left(\frac{-1}{n} + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}\right)}\right) \]

    4. add-sqr-sqrt 20.0%

       \[\leadsto 1 \cdot \left(\frac{1}{x} \cdot \left(\color{blue}{\sqrt{\frac{-1}{n}} \cdot \sqrt{\frac{-1}{n}}} + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}\right)\right) \]

    5. sqrt-unprod 37.0%

       \[\leadsto 1 \cdot \left(\frac{1}{x} \cdot \left(\color{blue}{\sqrt{\frac{-1}{n} \cdot \frac{-1}{n}}} + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}\right)\right) \]

    6. frac-times 37.0%

       \[\leadsto 1 \cdot \left(\frac{1}{x} \cdot \left(\sqrt{\color{blue}{\frac{-1 \cdot -1}{n \cdot n}}} + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}\right)\right) \]

    7. metadata-eval 37.0%

       \[\leadsto 1 \cdot \left(\frac{1}{x} \cdot \left(\sqrt{\frac{\color{blue}{1}}{n \cdot n}} + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}\right)\right) \]

    8. metadata-eval 37.0%

       \[\leadsto 1 \cdot \left(\frac{1}{x} \cdot \left(\sqrt{\frac{\color{blue}{1 \cdot 1}}{n \cdot n}} + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}\right)\right) \]

    9. frac-times 37.0%

       \[\leadsto 1 \cdot \left(\frac{1}{x} \cdot \left(\sqrt{\color{blue}{\frac{1}{n} \cdot \frac{1}{n}}} + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}\right)\right) \]

    10. sqrt-unprod 23.8%

        \[\leadsto 1 \cdot \left(\frac{1}{x} \cdot \left(\color{blue}{\sqrt{\frac{1}{n}} \cdot \sqrt{\frac{1}{n}}} + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}\right)\right) \]

    11. add-sqr-sqrt 50.8%

        \[\leadsto 1 \cdot \left(\frac{1}{x} \cdot \left(\color{blue}{\frac{1}{n}} + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}\right)\right) \]

    12. associate-/r* 50.8%

        \[\leadsto 1 \cdot \left(\frac{1}{x} \cdot \left(\frac{1}{n} + \frac{\color{blue}{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{n}} - \frac{0.5}{n}}{x}\right)\right) \]

    13. sub-div 50.8%

        \[\leadsto 1 \cdot \left(\frac{1}{x} \cdot \left(\frac{1}{n} + \frac{\color{blue}{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} - 0.5}{n}}}{x}\right)\right) \]

14. Applied egg-rr 50.8%

    \[\leadsto \color{blue}{1 \cdot \left(\frac{1}{x} \cdot \left(\frac{1}{n} + \frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} - 0.5}{n}}{x}\right)\right)} \]
15. Step-by-step derivation

    1. *-lft-identity 50.8%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{n} + \frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} - 0.5}{n}}{x}\right)} \]

    2. associate-*l/ 50.8%

       \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{1}{n} + \frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} - 0.5}{n}}{x}\right)}{x}} \]

    3. *-lft-identity 50.8%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + \frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} - 0.5}{n}}{x}}}{x} \]

    4. associate-/l/ 50.8%

       \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} - 0.5}{x \cdot n}}}{x} \]

    5. sub-neg 50.8%

       \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + \left(-0.5\right)}}{x \cdot n}}{x} \]

    6. metadata-eval 50.8%

       \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{\color{blue}{0.25 \cdot 1}}{x} + -0.3333333333333333}{x} + \left(-0.5\right)}{x \cdot n}}{x} \]

    7. associate-*r/ 50.8%

       \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\color{blue}{0.25 \cdot \frac{1}{x}} + -0.3333333333333333}{x} + \left(-0.5\right)}{x \cdot n}}{x} \]

    8. +-commutative 50.8%

       \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\color{blue}{-0.3333333333333333 + 0.25 \cdot \frac{1}{x}}}{x} + \left(-0.5\right)}{x \cdot n}}{x} \]

    9. associate-*r/ 50.8%

       \[\leadsto \frac{\frac{1}{n} + \frac{\frac{-0.3333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{x}}}{x} + \left(-0.5\right)}{x \cdot n}}{x} \]

    10. metadata-eval 50.8%

        \[\leadsto \frac{\frac{1}{n} + \frac{\frac{-0.3333333333333333 + \frac{\color{blue}{0.25}}{x}}{x} + \left(-0.5\right)}{x \cdot n}}{x} \]

    11. metadata-eval 50.8%

        \[\leadsto \frac{\frac{1}{n} + \frac{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x} + \color{blue}{-0.5}}{x \cdot n}}{x} \]

16. Simplified 50.8%

    \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x} + -0.5}{x \cdot n}}{x}} \]

17. Final simplification 50.8%

    \[\leadsto \frac{\frac{1}{n} + \frac{-0.5 + \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{n \cdot x}}{x} \]
18. Add Preprocessing

Alternative 17: 46.8% 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 54.0%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 57.4%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-define 57.4%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

5. Simplified 57.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

6. Taylor expanded in x around -inf 31.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
7. Step-by-step derivation

   1. associate-*r/ 31.1%

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}\right)}{x}} \]

8. Simplified 31.1%

   \[\leadsto \color{blue}{\frac{-\left(\frac{-\left(\frac{-\left(\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}\right)}{x} - \frac{0.5}{n}\right)}{x} - \frac{1}{n}\right)}{x}} \]

9. Taylor expanded in x around -inf 50.3%

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{0.5 \cdot \frac{1}{n} - 0.3333333333333333 \cdot \frac{1}{n \cdot x}}{x} + \frac{1}{n}}}{x} \]
10. Step-by-step derivation

    1. +-commutative 50.3%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + -1 \cdot \frac{0.5 \cdot \frac{1}{n} - 0.3333333333333333 \cdot \frac{1}{n \cdot x}}{x}}}{x} \]

    2. mul-1-neg 50.3%

       \[\leadsto \frac{\frac{1}{n} + \color{blue}{\left(-\frac{0.5 \cdot \frac{1}{n} - 0.3333333333333333 \cdot \frac{1}{n \cdot x}}{x}\right)}}{x} \]

    3. unsub-neg 50.3%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} - \frac{0.5 \cdot \frac{1}{n} - 0.3333333333333333 \cdot \frac{1}{n \cdot x}}{x}}}{x} \]

    4. sub-neg 50.3%

       \[\leadsto \frac{\frac{1}{n} - \frac{\color{blue}{0.5 \cdot \frac{1}{n} + \left(-0.3333333333333333 \cdot \frac{1}{n \cdot x}\right)}}{x}}{x} \]

    5. associate-*r/ 50.3%

       \[\leadsto \frac{\frac{1}{n} - \frac{\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.3333333333333333 \cdot \frac{1}{n \cdot x}\right)}{x}}{x} \]

    6. metadata-eval 50.3%

       \[\leadsto \frac{\frac{1}{n} - \frac{\frac{\color{blue}{0.5}}{n} + \left(-0.3333333333333333 \cdot \frac{1}{n \cdot x}\right)}{x}}{x} \]

    7. associate-*r/ 50.3%

       \[\leadsto \frac{\frac{1}{n} - \frac{\frac{0.5}{n} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{n \cdot x}}\right)}{x}}{x} \]

    8. metadata-eval 50.3%

       \[\leadsto \frac{\frac{1}{n} - \frac{\frac{0.5}{n} + \left(-\frac{\color{blue}{0.3333333333333333}}{n \cdot x}\right)}{x}}{x} \]

    9. *-commutative 50.3%

       \[\leadsto \frac{\frac{1}{n} - \frac{\frac{0.5}{n} + \left(-\frac{0.3333333333333333}{\color{blue}{x \cdot n}}\right)}{x}}{x} \]

    10. distribute-neg-frac 50.3%

        \[\leadsto \frac{\frac{1}{n} - \frac{\frac{0.5}{n} + \color{blue}{\frac{-0.3333333333333333}{x \cdot n}}}{x}}{x} \]

    11. metadata-eval 50.3%

        \[\leadsto \frac{\frac{1}{n} - \frac{\frac{0.5}{n} + \frac{\color{blue}{-0.3333333333333333}}{x \cdot n}}{x}}{x} \]

11. Simplified 50.3%

    \[\leadsto \frac{\color{blue}{\frac{1}{n} - \frac{\frac{0.5}{n} + \frac{-0.3333333333333333}{x \cdot n}}{x}}}{x} \]

12. Final simplification 50.3%

    \[\leadsto \frac{\frac{1}{n} - \frac{\frac{0.5}{n} + \frac{-0.3333333333333333}{n \cdot x}}{x}}{x} \]
13. Add Preprocessing

Alternative 18: 46.8% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x}}{n} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (/ (+ 1.0 (/ (- (* 0.3333333333333333 (/ 1.0 x)) 0.5) x)) x) n))

C

double code(double x, double n) {
	return ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 + (((0.3333333333333333d0 * (1.0d0 / x)) - 0.5d0) / x)) / x) / n
end function

Java

public static double code(double x, double n) {
	return ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n;
}

Python

def code(x, n):
	return ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 + Float64(Float64(Float64(0.3333333333333333 * Float64(1.0 / x)) - 0.5) / x)) / x) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n;
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 + N[(N[(N[(0.3333333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]

Derivation

1. Initial program 54.0%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 57.4%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-define 57.4%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

5. Simplified 57.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

6. Taylor expanded in x around -inf 50.3%

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]

7. Final simplification 50.3%

   \[\leadsto \frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x}}{n} \]
8. Add Preprocessing

Alternative 19: 41.2% 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 54.0%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 57.4%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-define 57.4%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

5. Simplified 57.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

6. Taylor expanded in x around inf 42.9%

   \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
7. Add Preprocessing

Alternative 20: 41.2% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n}}{x} \end{array} \]

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 54.0%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 57.4%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-define 57.4%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

5. Simplified 57.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

6. Taylor expanded in x around -inf 31.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
7. Step-by-step derivation

   1. associate-*r/ 31.1%

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}\right)}{x}} \]

8. Simplified 31.1%

   \[\leadsto \color{blue}{\frac{-\left(\frac{-\left(\frac{-\left(\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}\right)}{x} - \frac{0.5}{n}\right)}{x} - \frac{1}{n}\right)}{x}} \]

9. Taylor expanded in x around inf 42.9%

   \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]
10. Add Preprocessing

Alternative 21: 40.6% 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 54.0%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 57.4%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-define 57.4%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

5. Simplified 57.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

6. Taylor expanded in x around inf 41.7%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
7. Step-by-step derivation

   1. *-commutative 41.7%

      \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

8. Simplified 41.7%

   \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]

9. Final simplification 41.7%

   \[\leadsto \frac{1}{n \cdot x} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))