2nthrt (problem 3.4.6)

Percentage Accurate: 53.9% → 86.4%

Time: 24.3s

Alternatives: 19

Speedup: 2.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + x\right)\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.02:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{t_1}\right)}^{3}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\left(0.5 \cdot \frac{{t_0}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{t_0}^{4}}{{n}^{4}} + \left(\frac{-0.16666666666666666 \cdot {\log x}^{3} - -0.16666666666666666 \cdot {t_0}^{3}}{{n}^{3}} + \frac{t_0 - \log x}{n}\right)\right)\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + 0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (log (+ 1.0 x))) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -0.02)
     (- (pow (+ 1.0 x) (/ 1.0 n)) (pow (cbrt t_1) 3.0))
     (if (<= (/ 1.0 n) 1e-11)
       (-
        (+
         (* 0.5 (/ (pow t_0 2.0) (pow n 2.0)))
         (+
          (* 0.041666666666666664 (/ (pow t_0 4.0) (pow n 4.0)))
          (+
           (/
            (-
             (* -0.16666666666666666 (pow (log x) 3.0))
             (* -0.16666666666666666 (pow t_0 3.0)))
            (pow n 3.0))
           (/ (- t_0 (log x)) n))))
        (+
         (* 0.5 (/ (pow (log x) 2.0) (pow n 2.0)))
         (* 0.041666666666666664 (/ (pow (log x) 4.0) (pow n 4.0)))))
       (- (exp (/ (log1p x) n)) t_1)))))

C

double code(double x, double n) {
	double t_0 = log((1.0 + x));
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.02) {
		tmp = pow((1.0 + x), (1.0 / n)) - pow(cbrt(t_1), 3.0);
	} else if ((1.0 / n) <= 1e-11) {
		tmp = ((0.5 * (pow(t_0, 2.0) / pow(n, 2.0))) + ((0.041666666666666664 * (pow(t_0, 4.0) / pow(n, 4.0))) + ((((-0.16666666666666666 * pow(log(x), 3.0)) - (-0.16666666666666666 * pow(t_0, 3.0))) / pow(n, 3.0)) + ((t_0 - log(x)) / n)))) - ((0.5 * (pow(log(x), 2.0) / pow(n, 2.0))) + (0.041666666666666664 * (pow(log(x), 4.0) / pow(n, 4.0))));
	} else {
		tmp = exp((log1p(x) / n)) - t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log((1.0 + x));
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.02) {
		tmp = Math.pow((1.0 + x), (1.0 / n)) - Math.pow(Math.cbrt(t_1), 3.0);
	} else if ((1.0 / n) <= 1e-11) {
		tmp = ((0.5 * (Math.pow(t_0, 2.0) / Math.pow(n, 2.0))) + ((0.041666666666666664 * (Math.pow(t_0, 4.0) / Math.pow(n, 4.0))) + ((((-0.16666666666666666 * Math.pow(Math.log(x), 3.0)) - (-0.16666666666666666 * Math.pow(t_0, 3.0))) / Math.pow(n, 3.0)) + ((t_0 - Math.log(x)) / n)))) - ((0.5 * (Math.pow(Math.log(x), 2.0) / Math.pow(n, 2.0))) + (0.041666666666666664 * (Math.pow(Math.log(x), 4.0) / Math.pow(n, 4.0))));
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_1;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = log(Float64(1.0 + x))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.02)
		tmp = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - (cbrt(t_1) ^ 3.0));
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(Float64(Float64(0.5 * Float64((t_0 ^ 2.0) / (n ^ 2.0))) + Float64(Float64(0.041666666666666664 * Float64((t_0 ^ 4.0) / (n ^ 4.0))) + Float64(Float64(Float64(Float64(-0.16666666666666666 * (log(x) ^ 3.0)) - Float64(-0.16666666666666666 * (t_0 ^ 3.0))) / (n ^ 3.0)) + Float64(Float64(t_0 - log(x)) / n)))) - Float64(Float64(0.5 * Float64((log(x) ^ 2.0) / (n ^ 2.0))) + Float64(0.041666666666666664 * Float64((log(x) ^ 4.0) / (n ^ 4.0)))));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_1);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Log[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.02], N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[(N[(0.5 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.041666666666666664 * N[(N[Power[t$95$0, 4.0], $MachinePrecision] / N[Power[n, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(-0.16666666666666666 * N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] - N[(-0.16666666666666666 * N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[n, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[(N[Power[N[Log[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[n, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -0.0200000000000000004

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. unpow-prod-down 100.0%

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

      3. inv-pow 100.0%

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

      4. inv-pow 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

      3. pow-prod-down 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. inv-pow 100.0%

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

   5. Applied egg-rr 100.0%

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

   if -0.0200000000000000004 < (/.f64 1 n) < 9.99999999999999939e-12

   1. Initial program 30.6%

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

   2. Taylor expanded in n around -inf 80.1%

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

   if 9.99999999999999939e-12 < (/.f64 1 n)

   1. Initial program 43.3%

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

   2. Taylor expanded in n around 0 43.2%

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

      1. log1p-def 86.1%

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

      2. *-rgt-identity 86.1%

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

      3. associate-*r/ 86.0%

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

      4. unpow-1 86.0%

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

      5. exp-to-pow 86.0%

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

      6. /-rgt-identity 86.0%

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

      7. metadata-eval 86.0%

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

      8. associate-/l* 86.0%

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

      9. *-commutative 86.0%

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

      10. *-commutative 86.0%

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

      11. associate-/l* 86.0%

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

      12. metadata-eval 86.0%

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

      13. /-rgt-identity 86.0%

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

      14. unpow-1 86.0%

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

   4. Simplified 86.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.02:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{\log \left(1 + x\right)}^{4}}{{n}^{4}} + \left(\frac{-0.16666666666666666 \cdot {\log x}^{3} - -0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3}}{{n}^{3}} + \frac{\log \left(1 + x\right) - \log x}{n}\right)\right)\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + 0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2: 86.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.02:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{t_0}\right)}^{3}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{{n}^{3}}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -0.02)
     (- (pow (+ 1.0 x) (/ 1.0 n)) (pow (cbrt t_0) 3.0))
     (if (<= (/ 1.0 n) 1e-11)
       (+
        (+
         (fma 0.5 (/ (pow (log1p x) 2.0) (* n n)) (/ (- (log1p x) (log x)) n))
         (/
          (* 0.16666666666666666 (- (pow (log1p x) 3.0) (pow (log x) 3.0)))
          (pow n 3.0)))
        (* (/ (pow (log x) 2.0) (* n n)) -0.5))
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.02) {
		tmp = pow((1.0 + x), (1.0 / n)) - pow(cbrt(t_0), 3.0);
	} else if ((1.0 / n) <= 1e-11) {
		tmp = (fma(0.5, (pow(log1p(x), 2.0) / (n * n)), ((log1p(x) - log(x)) / n)) + ((0.16666666666666666 * (pow(log1p(x), 3.0) - pow(log(x), 3.0))) / pow(n, 3.0))) + ((pow(log(x), 2.0) / (n * n)) * -0.5);
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.02)
		tmp = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - (cbrt(t_0) ^ 3.0));
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(Float64(fma(0.5, Float64((log1p(x) ^ 2.0) / Float64(n * n)), Float64(Float64(log1p(x) - log(x)) / n)) + Float64(Float64(0.16666666666666666 * Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))) / (n ^ 3.0))) + Float64(Float64((log(x) ^ 2.0) / Float64(n * n)) * -0.5));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.02], N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(N[(0.16666666666666666 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[n, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -0.0200000000000000004

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. unpow-prod-down 100.0%

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

      3. inv-pow 100.0%

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

      4. inv-pow 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

      3. pow-prod-down 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. inv-pow 100.0%

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

   5. Applied egg-rr 100.0%

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

   if -0.0200000000000000004 < (/.f64 1 n) < 9.99999999999999939e-12

   1. Initial program 30.6%

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

   2. Taylor expanded in n around -inf 80.1%

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

      1. sub-neg 80.1%

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

   4. Simplified 80.1%

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

   if 9.99999999999999939e-12 < (/.f64 1 n)

   1. Initial program 43.3%

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

   2. Taylor expanded in n around 0 43.2%

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

      1. log1p-def 86.1%

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

      2. *-rgt-identity 86.1%

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

      3. associate-*r/ 86.0%

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

      4. unpow-1 86.0%

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

      5. exp-to-pow 86.0%

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

      6. /-rgt-identity 86.0%

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

      7. metadata-eval 86.0%

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

      8. associate-/l* 86.0%

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

      9. *-commutative 86.0%

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

      10. *-commutative 86.0%

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

      11. associate-/l* 86.0%

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

      12. metadata-eval 86.0%

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

      13. /-rgt-identity 86.0%

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

      14. unpow-1 86.0%

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

   4. Simplified 86.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.02:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{{n}^{3}}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 3: 86.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{t_0}\right)}^{3}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-7)
     (- (pow (+ 1.0 x) (/ 1.0 n)) (pow (cbrt t_0) 3.0))
     (if (<= (/ 1.0 n) 1e-11)
       (+
        (fma 0.5 (/ (pow (log1p x) 2.0) (* n n)) (/ (- (log1p x) (log x)) n))
        (* (/ (pow (log x) 2.0) (* n n)) -0.5))
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-7) {
		tmp = pow((1.0 + x), (1.0 / n)) - pow(cbrt(t_0), 3.0);
	} else if ((1.0 / n) <= 1e-11) {
		tmp = fma(0.5, (pow(log1p(x), 2.0) / (n * n)), ((log1p(x) - log(x)) / n)) + ((pow(log(x), 2.0) / (n * n)) * -0.5);
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-7)
		tmp = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - (cbrt(t_0) ^ 3.0));
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(fma(0.5, Float64((log1p(x) ^ 2.0) / Float64(n * n)), Float64(Float64(log1p(x) - log(x)) / n)) + Float64(Float64((log(x) ^ 2.0) / Float64(n * n)) * -0.5));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-7], N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -9.9999999999999995e-8

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 99.7%

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

      2. unpow-prod-down 99.7%

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

      3. inv-pow 99.7%

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

      4. inv-pow 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. add-cube-cbrt 99.7%

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

      2. pow3 99.7%

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

      3. pow-prod-down 99.8%

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

      4. add-sqr-sqrt 99.8%

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

      5. inv-pow 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -9.9999999999999995e-8 < (/.f64 1 n) < 9.99999999999999939e-12

   1. Initial program 30.2%

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

   2. Taylor expanded in n around inf 80.0%

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

      1. associate--r+ 72.2%

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

      2. sub-neg 72.2%

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

   4. Simplified 80.0%

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

   if 9.99999999999999939e-12 < (/.f64 1 n)

   1. Initial program 43.3%

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

   2. Taylor expanded in n around 0 43.2%

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

      1. log1p-def 86.1%

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

      2. *-rgt-identity 86.1%

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

      3. associate-*r/ 86.0%

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

      4. unpow-1 86.0%

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

      5. exp-to-pow 86.0%

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

      6. /-rgt-identity 86.0%

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

      7. metadata-eval 86.0%

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

      8. associate-/l* 86.0%

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

      9. *-commutative 86.0%

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

      10. *-commutative 86.0%

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

      11. associate-/l* 86.0%

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

      12. metadata-eval 86.0%

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

      13. /-rgt-identity 86.0%

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

      14. unpow-1 86.0%

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

   4. Simplified 86.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 4: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{t_0}\right)}^{3}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-7)
     (- (pow (+ 1.0 x) (/ 1.0 n)) (pow (cbrt t_0) 3.0))
     (if (<= (/ 1.0 n) 1e-11)
       (/ (log (/ (+ 1.0 x) x)) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-7) {
		tmp = pow((1.0 + x), (1.0 / n)) - pow(cbrt(t_0), 3.0);
	} else if ((1.0 / n) <= 1e-11) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-7) {
		tmp = Math.pow((1.0 + x), (1.0 / n)) - Math.pow(Math.cbrt(t_0), 3.0);
	} else if ((1.0 / n) <= 1e-11) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-7)
		tmp = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - (cbrt(t_0) ^ 3.0));
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-7], N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -9.9999999999999995e-8

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 99.7%

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

      2. unpow-prod-down 99.7%

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

      3. inv-pow 99.7%

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

      4. inv-pow 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. add-cube-cbrt 99.7%

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

      2. pow3 99.7%

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

      3. pow-prod-down 99.8%

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

      4. add-sqr-sqrt 99.8%

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

      5. inv-pow 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -9.9999999999999995e-8 < (/.f64 1 n) < 9.99999999999999939e-12

   1. Initial program 30.2%

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

   2. Taylor expanded in n around inf 79.1%

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

      1. log1p-def 79.1%

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

   4. Simplified 79.1%

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

      1. log1p-udef 79.1%

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

      2. +-commutative 79.1%

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

      3. diff-log 79.4%

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

   6. Applied egg-rr 79.4%

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

   if 9.99999999999999939e-12 < (/.f64 1 n)

   1. Initial program 43.3%

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

   2. Taylor expanded in n around 0 43.2%

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

      1. log1p-def 86.1%

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

      2. *-rgt-identity 86.1%

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

      3. associate-*r/ 86.0%

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

      4. unpow-1 86.0%

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

      5. exp-to-pow 86.0%

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

      6. /-rgt-identity 86.0%

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

      7. metadata-eval 86.0%

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

      8. associate-/l* 86.0%

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

      9. *-commutative 86.0%

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

      10. *-commutative 86.0%

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

      11. associate-/l* 86.0%

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

      12. metadata-eval 86.0%

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

      13. /-rgt-identity 86.0%

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

      14. unpow-1 86.0%

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

   4. Simplified 86.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 5: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;e^{\frac{x}{n}} - t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-7)
     (- (exp (/ x n)) t_0)
     (if (<= (/ 1.0 n) 1e-11)
       (/ (log (/ (+ 1.0 x) x)) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-7) {
		tmp = exp((x / n)) - t_0;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-7) {
		tmp = Math.exp((x / n)) - t_0;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-7:
		tmp = math.exp((x / n)) - t_0
	elif (1.0 / n) <= 1e-11:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-7)
		tmp = Float64(exp(Float64(x / n)) - t_0);
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-7], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -9.9999999999999995e-8

   1. Initial program 99.7%

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

   2. Taylor expanded in n around 0 99.7%

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

      1. log1p-def 99.7%

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

      2. *-rgt-identity 99.7%

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

      3. associate-*r/ 99.7%

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

      4. unpow-1 99.7%

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

      5. exp-to-pow 99.7%

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

      6. /-rgt-identity 99.7%

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

      7. metadata-eval 99.7%

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

      8. associate-/l* 99.7%

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

      9. *-commutative 99.7%

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

      10. *-commutative 99.7%

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

      11. associate-/l* 99.7%

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

      12. metadata-eval 99.7%

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

      13. /-rgt-identity 99.7%

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

      14. unpow-1 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

   if -9.9999999999999995e-8 < (/.f64 1 n) < 9.99999999999999939e-12

   1. Initial program 30.2%

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

   2. Taylor expanded in n around inf 79.1%

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

      1. log1p-def 79.1%

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

   4. Simplified 79.1%

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

      1. log1p-udef 79.1%

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

      2. +-commutative 79.1%

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

      3. diff-log 79.4%

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

   6. Applied egg-rr 79.4%

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

   if 9.99999999999999939e-12 < (/.f64 1 n)

   1. Initial program 43.3%

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

   2. Taylor expanded in n around 0 43.2%

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

      1. log1p-def 86.1%

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

      2. *-rgt-identity 86.1%

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

      3. associate-*r/ 86.0%

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

      4. unpow-1 86.0%

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

      5. exp-to-pow 86.0%

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

      6. /-rgt-identity 86.0%

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

      7. metadata-eval 86.0%

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

      8. associate-/l* 86.0%

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

      9. *-commutative 86.0%

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

      10. *-commutative 86.0%

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

      11. associate-/l* 86.0%

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

      12. metadata-eval 86.0%

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

      13. /-rgt-identity 86.0%

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

      14. unpow-1 86.0%

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

   4. Simplified 86.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 6: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-7} \lor \neg \left(\frac{1}{n} \leq 4 \cdot 10^{-5}\right):\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= (/ 1.0 n) -1e-7) (not (<= (/ 1.0 n) 4e-5)))
   (- (exp (/ x n)) (pow x (/ 1.0 n)))
   (/ (log (/ (+ 1.0 x) x)) n)))

C

double code(double x, double n) {
	double tmp;
	if (((1.0 / n) <= -1e-7) || !((1.0 / n) <= 4e-5)) {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = log(((1.0 + x) / x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (((1.0d0 / n) <= (-1d-7)) .or. (.not. ((1.0d0 / n) <= 4d-5))) then
        tmp = exp((x / n)) - (x ** (1.0d0 / n))
    else
        tmp = log(((1.0d0 + x) / x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (((1.0 / n) <= -1e-7) || !((1.0 / n) <= 4e-5)) {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log(((1.0 + x) / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if ((1.0 / n) <= -1e-7) or not ((1.0 / n) <= 4e-5):
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.log(((1.0 + x) / x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((Float64(1.0 / n) <= -1e-7) || !(Float64(1.0 / n) <= 4e-5))
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (((1.0 / n) <= -1e-7) || ~(((1.0 / n) <= 4e-5)))
		tmp = exp((x / n)) - (x ^ (1.0 / n));
	else
		tmp = log(((1.0 + x) / x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-7], N[Not[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-5]], $MachinePrecision]], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 n) < -9.9999999999999995e-8 or 4.00000000000000033e-5 < (/.f64 1 n)

   1. Initial program 78.3%

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

   2. Taylor expanded in n around 0 78.2%

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

      1. log1p-def 96.4%

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

      2. *-rgt-identity 96.4%

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

      3. associate-*r/ 96.4%

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

      4. unpow-1 96.4%

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

      5. exp-to-pow 96.4%

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

      6. /-rgt-identity 96.4%

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

      7. metadata-eval 96.4%

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

      8. associate-/l* 96.4%

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

      9. *-commutative 96.4%

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

      10. *-commutative 96.4%

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

      11. associate-/l* 96.4%

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

      12. metadata-eval 96.4%

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

      13. /-rgt-identity 96.4%

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

      14. unpow-1 96.4%

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

   4. Simplified 96.4%

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

   5. Taylor expanded in x around 0 96.2%

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

   if -9.9999999999999995e-8 < (/.f64 1 n) < 4.00000000000000033e-5

   1. Initial program 29.7%

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

   2. Taylor expanded in n around inf 77.6%

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

      1. log1p-def 77.6%

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

   4. Simplified 77.6%

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

      1. log1p-udef 77.6%

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

      2. +-commutative 77.6%

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

      3. diff-log 77.8%

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

   6. Applied egg-rr 77.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-7} \lor \neg \left(\frac{1}{n} \leq 4 \cdot 10^{-5}\right):\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \end{array} \]

Alternative 7: 81.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{1}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -10000:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+91}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{t_1 \cdot \frac{t_1}{n \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ 1.0 (* n x))))
   (if (<= (/ 1.0 n) -10000.0)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 4e-5)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e+91)
         (- (+ 1.0 (/ x n)) t_0)
         (cbrt (* t_1 (/ t_1 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = 1.0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -10000.0) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 4e-5) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+91) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = cbrt((t_1 * (t_1 / (n * x))));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = 1.0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -10000.0) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 4e-5) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+91) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.cbrt((t_1 * (t_1 / (n * x))));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(1.0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -10000.0)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 4e-5)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+91)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = cbrt(Float64(t_1 * Float64(t_1 / Float64(n * x))));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -10000.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-5], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+91], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Power[N[(t$95$1 * N[(t$95$1 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1e4

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-rgt-identity 100.0%

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

      8. associate-*r/ 100.0%

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

      9. unpow-1 100.0%

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

      10. exp-to-pow 100.0%

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

      11. unpow-1 100.0%

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

      12. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   if -1e4 < (/.f64 1 n) < 4.00000000000000033e-5

   1. Initial program 30.5%

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

   2. Taylor expanded in n around inf 76.9%

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

      1. log1p-def 76.9%

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

   4. Simplified 76.9%

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

      1. log1p-udef 76.9%

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

      2. +-commutative 76.9%

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

      3. diff-log 77.1%

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

   6. Applied egg-rr 77.1%

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

   if 4.00000000000000033e-5 < (/.f64 1 n) < 2.00000000000000016e91

   1. Initial program 74.0%

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

   2. Taylor expanded in x around 0 72.8%

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

   if 2.00000000000000016e91 < (/.f64 1 n)

   1. Initial program 27.2%

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

   2. Taylor expanded in n around inf 10.1%

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

      1. log1p-def 10.1%

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

   4. Simplified 10.1%

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

   5. Taylor expanded in x around inf 57.6%

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

      1. *-commutative 57.6%

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

   7. Simplified 57.6%

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

      1. add-cbrt-cube 75.0%

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

   9. Applied egg-rr 75.0%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1}{x \cdot n} \cdot \frac{1}{x \cdot n}\right) \cdot \frac{1}{x \cdot n}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 75.0%

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

       2. *-rgt-identity 75.0%

          \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{1}{x \cdot n} \cdot \frac{1}{x \cdot n}}}{x \cdot n}} \]

       3. associate-*r/ 75.0%

          \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{x \cdot n} \cdot \frac{\frac{1}{x \cdot n}}{x \cdot n}}} \]

   11. Simplified 75.0%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+91}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{n \cdot x} \cdot \frac{\frac{1}{n \cdot x}}{n \cdot x}}\\ \end{array} \]

Alternative 8: 81.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -10000:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+204}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -10000.0)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 4e-5)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e+204)
         (- (+ 1.0 (/ x n)) t_0)
         (/ 0.3333333333333333 (* n (pow x 3.0))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -10000.0) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 4e-5) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+204) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-10000.0d0)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 4d-5) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+204) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -10000.0) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 4e-5) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+204) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -10000.0:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 4e-5:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+204:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -10000.0)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 4e-5)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+204)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -10000.0)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 4e-5)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+204)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -10000.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-5], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+204], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1e4

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-rgt-identity 100.0%

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

      8. associate-*r/ 100.0%

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

      9. unpow-1 100.0%

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

      10. exp-to-pow 100.0%

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

      11. unpow-1 100.0%

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

      12. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   if -1e4 < (/.f64 1 n) < 4.00000000000000033e-5

   1. Initial program 30.5%

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

   2. Taylor expanded in n around inf 76.9%

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

      1. log1p-def 76.9%

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

   4. Simplified 76.9%

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

      1. log1p-udef 76.9%

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

      2. +-commutative 76.9%

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

      3. diff-log 77.1%

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

   6. Applied egg-rr 77.1%

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

   if 4.00000000000000033e-5 < (/.f64 1 n) < 1.99999999999999998e204

   1. Initial program 69.8%

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

   2. Taylor expanded in x around 0 62.7%

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

   if 1.99999999999999998e204 < (/.f64 1 n)

   1. Initial program 13.3%

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

   2. Taylor expanded in n around inf 12.4%

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

      1. log1p-def 12.4%

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

   4. Simplified 12.4%

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

   5. Taylor expanded in x around inf 26.3%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate--l+ 26.3%

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

      2. associate-*r/ 26.3%

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

      3. metadata-eval 26.3%

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

      4. associate-*r/ 26.3%

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

      5. metadata-eval 26.3%

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

      6. unpow2 26.3%

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

   7. Simplified 26.3%

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

   8. Taylor expanded in x around 0 79.7%

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

      1. *-commutative 79.7%

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

   10. Simplified 79.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+204}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \]

Alternative 9: 73.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{if}\;\frac{1}{n} \leq -10000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+91}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 0.3333333333333333 (* n (pow x 3.0)))))
   (if (<= (/ 1.0 n) -10000.0)
     t_0
     (if (<= (/ 1.0 n) 4e-5)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e+91) (- 1.0 (pow x (/ 1.0 n))) t_0)))))

C

double code(double x, double n) {
	double t_0 = 0.3333333333333333 / (n * pow(x, 3.0));
	double tmp;
	if ((1.0 / n) <= -10000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-5) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+91) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    if ((1.0d0 / n) <= (-10000.0d0)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 4d-5) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+91) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	double tmp;
	if ((1.0 / n) <= -10000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-5) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+91) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 0.3333333333333333 / (n * math.pow(x, 3.0))
	tmp = 0
	if (1.0 / n) <= -10000.0:
		tmp = t_0
	elif (1.0 / n) <= 4e-5:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+91:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -10000.0)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 4e-5)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+91)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 0.3333333333333333 / (n * (x ^ 3.0));
	tmp = 0.0;
	if ((1.0 / n) <= -10000.0)
		tmp = t_0;
	elseif ((1.0 / n) <= 4e-5)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+91)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -10000.0], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-5], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+91], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -1e4 or 2.00000000000000016e91 < (/.f64 1 n)

   1. Initial program 78.9%

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

   2. Taylor expanded in n around inf 40.1%

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

      1. log1p-def 40.1%

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

   4. Simplified 40.1%

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

   5. Taylor expanded in x around inf 27.4%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate--l+ 27.4%

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

      2. associate-*r/ 27.4%

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

      3. metadata-eval 27.4%

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

      4. associate-*r/ 27.4%

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

      5. metadata-eval 27.4%

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

      6. unpow2 27.4%

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

   7. Simplified 27.4%

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

   8. Taylor expanded in x around 0 77.3%

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

      1. *-commutative 77.3%

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

   10. Simplified 77.3%

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

   if -1e4 < (/.f64 1 n) < 4.00000000000000033e-5

   1. Initial program 30.5%

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

   2. Taylor expanded in n around inf 76.9%

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

      1. log1p-def 76.9%

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

   4. Simplified 76.9%

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

      1. log1p-udef 76.9%

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

      2. +-commutative 76.9%

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

      3. diff-log 77.1%

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

   6. Applied egg-rr 77.1%

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

   if 4.00000000000000033e-5 < (/.f64 1 n) < 2.00000000000000016e91

   1. Initial program 74.0%

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

   2. Taylor expanded in x around 0 72.2%

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

      1. *-rgt-identity 72.2%

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

      2. associate-*r/ 72.2%

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

      3. unpow-1 72.2%

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

      4. exp-to-pow 72.2%

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

      5. unpow-1 72.2%

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

   4. Simplified 72.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10000:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+91}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \]

Alternative 10: 80.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -10000:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+91}:\\ \;\;\;\;1 - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -10000.0)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 4e-5)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e+91)
         (- 1.0 t_0)
         (/ 0.3333333333333333 (* n (pow x 3.0))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -10000.0) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 4e-5) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+91) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-10000.0d0)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 4d-5) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+91) then
        tmp = 1.0d0 - t_0
    else
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -10000.0) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 4e-5) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+91) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -10000.0:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 4e-5:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+91:
		tmp = 1.0 - t_0
	else:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -10000.0)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 4e-5)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+91)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -10000.0)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 4e-5)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+91)
		tmp = 1.0 - t_0;
	else
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -10000.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-5], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+91], N[(1.0 - t$95$0), $MachinePrecision], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1e4

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-rgt-identity 100.0%

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

      8. associate-*r/ 100.0%

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

      9. unpow-1 100.0%

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

      10. exp-to-pow 100.0%

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

      11. unpow-1 100.0%

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

      12. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   if -1e4 < (/.f64 1 n) < 4.00000000000000033e-5

   1. Initial program 30.5%

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

   2. Taylor expanded in n around inf 76.9%

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

      1. log1p-def 76.9%

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

   4. Simplified 76.9%

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

      1. log1p-udef 76.9%

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

      2. +-commutative 76.9%

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

      3. diff-log 77.1%

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

   6. Applied egg-rr 77.1%

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

   if 4.00000000000000033e-5 < (/.f64 1 n) < 2.00000000000000016e91

   1. Initial program 74.0%

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

   2. Taylor expanded in x around 0 72.2%

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

      1. *-rgt-identity 72.2%

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

      2. associate-*r/ 72.2%

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

      3. unpow-1 72.2%

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

      4. exp-to-pow 72.2%

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

      5. unpow-1 72.2%

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

   4. Simplified 72.2%

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

   if 2.00000000000000016e91 < (/.f64 1 n)

   1. Initial program 27.2%

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

   2. Taylor expanded in n around inf 10.1%

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

      1. log1p-def 10.1%

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

   4. Simplified 10.1%

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

   5. Taylor expanded in x around inf 30.0%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate--l+ 30.0%

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

      2. associate-*r/ 30.0%

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

      3. metadata-eval 30.0%

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

      4. associate-*r/ 30.0%

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

      5. metadata-eval 30.0%

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

      6. unpow2 30.0%

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

   7. Simplified 30.0%

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

   8. Taylor expanded in x around 0 67.9%

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

      1. *-commutative 67.9%

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

   10. Simplified 67.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+91}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \]

Alternative 11: 59.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-87}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-50}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\frac{1}{n} \cdot \left(x - \log x\right)\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.8e-87)
   (/ (- (log x)) n)
   (if (<= x 1.75e-50)
     (/ 0.3333333333333333 (* n (pow x 3.0)))
     (if (<= x 0.96)
       (* (/ 1.0 n) (- x (log x)))
       (if (<= x 6.2e+82) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.8e-87) {
		tmp = -log(x) / n;
	} else if (x <= 1.75e-50) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if (x <= 0.96) {
		tmp = (1.0 / n) * (x - log(x));
	} else if (x <= 6.2e+82) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.8d-87) then
        tmp = -log(x) / n
    else if (x <= 1.75d-50) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if (x <= 0.96d0) then
        tmp = (1.0d0 / n) * (x - log(x))
    else if (x <= 6.2d+82) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.8e-87) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.75e-50) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if (x <= 0.96) {
		tmp = (1.0 / n) * (x - Math.log(x));
	} else if (x <= 6.2e+82) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.8e-87:
		tmp = -math.log(x) / n
	elif x <= 1.75e-50:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif x <= 0.96:
		tmp = (1.0 / n) * (x - math.log(x))
	elif x <= 6.2e+82:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.8e-87)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.75e-50)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (x <= 0.96)
		tmp = Float64(Float64(1.0 / n) * Float64(x - log(x)));
	elseif (x <= 6.2e+82)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.8e-87)
		tmp = -log(x) / n;
	elseif (x <= 1.75e-50)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif (x <= 0.96)
		tmp = (1.0 / n) * (x - log(x));
	elseif (x <= 6.2e+82)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.8e-87], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.75e-50], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.96], N[(N[(1.0 / n), $MachinePrecision] * N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.2e+82], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.79999999999999996e-87

   1. Initial program 36.9%

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

   2. Taylor expanded in x around 0 36.8%

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

      1. *-rgt-identity 36.8%

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

      2. associate-*r/ 36.8%

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

      3. unpow-1 36.8%

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

      4. exp-to-pow 36.9%

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

      5. unpow-1 36.9%

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

   4. Simplified 36.9%

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

   5. Taylor expanded in n around inf 56.9%

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

      1. neg-mul-1 56.9%

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

      2. distribute-neg-frac 56.9%

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

   7. Simplified 56.9%

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

   if 1.79999999999999996e-87 < x < 1.74999999999999998e-50

   1. Initial program 67.8%

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

   2. Taylor expanded in n around inf 14.0%

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

      1. log1p-def 14.0%

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

   4. Simplified 14.0%

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

   5. Taylor expanded in x around inf 83.9%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate--l+ 83.9%

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

      2. associate-*r/ 83.9%

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

      3. metadata-eval 83.9%

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

      4. associate-*r/ 83.9%

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

      5. metadata-eval 83.9%

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

      6. unpow2 83.9%

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

   7. Simplified 83.9%

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

   8. Taylor expanded in x around 0 83.9%

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

      1. *-commutative 83.9%

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

   10. Simplified 83.9%

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

   if 1.74999999999999998e-50 < x < 0.95999999999999996

   1. Initial program 43.2%

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

   2. Taylor expanded in n around inf 55.7%

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

      1. log1p-def 55.7%

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

   4. Simplified 55.7%

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

      1. log1p-udef 55.7%

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

      2. +-commutative 55.7%

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

      3. diff-log 55.7%

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

   6. Applied egg-rr 55.7%

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

      1. div-inv 55.9%

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

   8. Applied egg-rr 55.9%

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

   9. Taylor expanded in x around 0 55.5%

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

       1. mul-1-neg 55.5%

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

       2. unsub-neg 55.5%

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

   11. Simplified 55.5%

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

   if 0.95999999999999996 < x < 6.20000000000000065e82

   1. Initial program 40.2%

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

   2. Taylor expanded in n around inf 40.2%

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

      1. log1p-def 40.2%

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

   4. Simplified 40.2%

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

   5. Taylor expanded in x around inf 67.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate-*r/ 67.2%

         \[\leadsto \frac{\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{n} \]

      2. metadata-eval 67.2%

         \[\leadsto \frac{\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}}{n} \]

      3. unpow2 67.2%

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

   7. Simplified 67.2%

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

   if 6.20000000000000065e82 < x

   1. Initial program 76.4%

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

   2. Taylor expanded in n around inf 76.5%

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

      1. log1p-def 76.5%

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

   4. Simplified 76.5%

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

      1. log1p-udef 76.5%

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

      2. +-commutative 76.5%

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

      3. diff-log 76.5%

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

   6. Applied egg-rr 76.5%

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

   7. Taylor expanded in x around inf 76.5%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-87}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-50}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\frac{1}{n} \cdot \left(x - \log x\right)\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 12: 59.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{-87}:\\ \;\;\;\;\frac{1}{0.5 - \frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-50}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\frac{1}{n} \cdot \left(x - \log x\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.75e-87)
   (/ 1.0 (- 0.5 (/ n (log x))))
   (if (<= x 1.55e-50)
     (/ 0.3333333333333333 (* n (pow x 3.0)))
     (if (<= x 0.96)
       (* (/ 1.0 n) (- x (log x)))
       (if (<= x 3.1e+88) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.75e-87) {
		tmp = 1.0 / (0.5 - (n / log(x)));
	} else if (x <= 1.55e-50) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if (x <= 0.96) {
		tmp = (1.0 / n) * (x - log(x));
	} else if (x <= 3.1e+88) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.75d-87) then
        tmp = 1.0d0 / (0.5d0 - (n / log(x)))
    else if (x <= 1.55d-50) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if (x <= 0.96d0) then
        tmp = (1.0d0 / n) * (x - log(x))
    else if (x <= 3.1d+88) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.75e-87) {
		tmp = 1.0 / (0.5 - (n / Math.log(x)));
	} else if (x <= 1.55e-50) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if (x <= 0.96) {
		tmp = (1.0 / n) * (x - Math.log(x));
	} else if (x <= 3.1e+88) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.75e-87:
		tmp = 1.0 / (0.5 - (n / math.log(x)))
	elif x <= 1.55e-50:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif x <= 0.96:
		tmp = (1.0 / n) * (x - math.log(x))
	elif x <= 3.1e+88:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.75e-87)
		tmp = Float64(1.0 / Float64(0.5 - Float64(n / log(x))));
	elseif (x <= 1.55e-50)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (x <= 0.96)
		tmp = Float64(Float64(1.0 / n) * Float64(x - log(x)));
	elseif (x <= 3.1e+88)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.75e-87)
		tmp = 1.0 / (0.5 - (n / log(x)));
	elseif (x <= 1.55e-50)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif (x <= 0.96)
		tmp = (1.0 / n) * (x - log(x));
	elseif (x <= 3.1e+88)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.75e-87], N[(1.0 / N[(0.5 - N[(n / N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e-50], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.96], N[(N[(1.0 / n), $MachinePrecision] * N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1e+88], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.75000000000000006e-87

   1. Initial program 36.9%

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

   2. Taylor expanded in x around 0 36.8%

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

      1. *-rgt-identity 36.8%

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

      2. associate-*r/ 36.8%

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

      3. unpow-1 36.8%

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

      4. exp-to-pow 36.9%

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

      5. unpow-1 36.9%

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

   4. Simplified 36.9%

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

      1. flip-- 19.5%

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

      2. clear-num 19.5%

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

      3. metadata-eval 19.5%

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

      4. inv-pow 19.5%

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

      5. inv-pow 19.5%

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

      6. pow-prod-down 16.5%

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

      7. pow2 16.5%

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

      8. pow-unpow 19.5%

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

      9. inv-pow 19.5%

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

      10. un-div-inv 19.5%

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

   6. Applied egg-rr 19.5%

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

   7. Taylor expanded in n around inf 57.4%

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

      1. mul-1-neg 57.4%

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

      2. unsub-neg 57.4%

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

   9. Simplified 57.4%

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

   if 1.75000000000000006e-87 < x < 1.5500000000000001e-50

   1. Initial program 67.8%

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

   2. Taylor expanded in n around inf 14.0%

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

      1. log1p-def 14.0%

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

   4. Simplified 14.0%

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

   5. Taylor expanded in x around inf 83.9%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate--l+ 83.9%

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

      2. associate-*r/ 83.9%

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

      3. metadata-eval 83.9%

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

      4. associate-*r/ 83.9%

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

      5. metadata-eval 83.9%

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

      6. unpow2 83.9%

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

   7. Simplified 83.9%

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

   8. Taylor expanded in x around 0 83.9%

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

      1. *-commutative 83.9%

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

   10. Simplified 83.9%

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

   if 1.5500000000000001e-50 < x < 0.95999999999999996

   1. Initial program 43.2%

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

   2. Taylor expanded in n around inf 55.7%

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

      1. log1p-def 55.7%

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

   4. Simplified 55.7%

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

      1. log1p-udef 55.7%

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

      2. +-commutative 55.7%

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

      3. diff-log 55.7%

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

   6. Applied egg-rr 55.7%

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

      1. div-inv 55.9%

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

   8. Applied egg-rr 55.9%

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

   9. Taylor expanded in x around 0 55.5%

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

       1. mul-1-neg 55.5%

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

       2. unsub-neg 55.5%

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

   11. Simplified 55.5%

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

   if 0.95999999999999996 < x < 3.1000000000000001e88

   1. Initial program 40.2%

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

   2. Taylor expanded in n around inf 40.2%

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

      1. log1p-def 40.2%

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

   4. Simplified 40.2%

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

   5. Taylor expanded in x around inf 67.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate-*r/ 67.2%

         \[\leadsto \frac{\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{n} \]

      2. metadata-eval 67.2%

         \[\leadsto \frac{\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}}{n} \]

      3. unpow2 67.2%

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

   7. Simplified 67.2%

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

   if 3.1000000000000001e88 < x

   1. Initial program 76.4%

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

   2. Taylor expanded in n around inf 76.5%

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

      1. log1p-def 76.5%

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

   4. Simplified 76.5%

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

      1. log1p-udef 76.5%

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

      2. +-commutative 76.5%

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

      3. diff-log 76.5%

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

   6. Applied egg-rr 76.5%

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

   7. Taylor expanded in x around inf 76.5%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{-87}:\\ \;\;\;\;\frac{1}{0.5 - \frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-50}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\frac{1}{n} \cdot \left(x - \log x\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 13: 59.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-80}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-56}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.1e-80)
   (/ (- (log x)) n)
   (if (<= x 3e-56)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 0.96)
       (/ (- x (log x)) n)
       (if (<= x 5.9e+88) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.1e-80) {
		tmp = -log(x) / n;
	} else if (x <= 3e-56) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.96) {
		tmp = (x - log(x)) / n;
	} else if (x <= 5.9e+88) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.1d-80) then
        tmp = -log(x) / n
    else if (x <= 3d-56) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.96d0) then
        tmp = (x - log(x)) / n
    else if (x <= 5.9d+88) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.1e-80) {
		tmp = -Math.log(x) / n;
	} else if (x <= 3e-56) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.96) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 5.9e+88) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.1e-80:
		tmp = -math.log(x) / n
	elif x <= 3e-56:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.96:
		tmp = (x - math.log(x)) / n
	elif x <= 5.9e+88:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.1e-80)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 3e-56)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.96)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 5.9e+88)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.1e-80)
		tmp = -log(x) / n;
	elseif (x <= 3e-56)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.96)
		tmp = (x - log(x)) / n;
	elseif (x <= 5.9e+88)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.1e-80], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 3e-56], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.96], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 5.9e+88], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.10000000000000005e-80

   1. Initial program 36.6%

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

   2. Taylor expanded in x around 0 36.6%

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

      1. *-rgt-identity 36.6%

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

      2. associate-*r/ 36.6%

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

      3. unpow-1 36.6%

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

      4. exp-to-pow 36.6%

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

      5. unpow-1 36.6%

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

   4. Simplified 36.6%

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

   5. Taylor expanded in n around inf 56.5%

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

      1. neg-mul-1 56.5%

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

      2. distribute-neg-frac 56.5%

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

   7. Simplified 56.5%

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

   if 1.10000000000000005e-80 < x < 2.99999999999999989e-56

   1. Initial program 80.8%

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

   2. Taylor expanded in x around 0 80.8%

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

      1. *-rgt-identity 80.8%

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

      2. associate-*r/ 80.8%

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

      3. unpow-1 80.8%

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

      4. exp-to-pow 80.8%

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

      5. unpow-1 80.8%

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

   4. Simplified 80.8%

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

   if 2.99999999999999989e-56 < x < 0.95999999999999996

   1. Initial program 40.7%

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

   2. Taylor expanded in n around inf 52.6%

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

      1. log1p-def 52.6%

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

   4. Simplified 52.6%

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

   5. Taylor expanded in x around 0 52.2%

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

      1. neg-mul-1 52.2%

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

      2. unsub-neg 52.2%

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

   7. Simplified 52.2%

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

   if 0.95999999999999996 < x < 5.89999999999999967e88

   1. Initial program 40.2%

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

   2. Taylor expanded in n around inf 40.2%

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

      1. log1p-def 40.2%

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

   4. Simplified 40.2%

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

   5. Taylor expanded in x around inf 67.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate-*r/ 67.2%

         \[\leadsto \frac{\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{n} \]

      2. metadata-eval 67.2%

         \[\leadsto \frac{\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}}{n} \]

      3. unpow2 67.2%

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

   7. Simplified 67.2%

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

   if 5.89999999999999967e88 < x

   1. Initial program 76.4%

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

   2. Taylor expanded in n around inf 76.5%

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

      1. log1p-def 76.5%

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

   4. Simplified 76.5%

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

      1. log1p-udef 76.5%

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

      2. +-commutative 76.5%

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

      3. diff-log 76.5%

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

   6. Applied egg-rr 76.5%

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

   7. Taylor expanded in x around inf 76.5%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-80}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-56}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 14: 59.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{-87}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-50}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.75e-87)
   (/ (- (log x)) n)
   (if (<= x 1.55e-50)
     (/ 0.3333333333333333 (* n (pow x 3.0)))
     (if (<= x 0.98)
       (/ (- x (log x)) n)
       (if (<= x 5.1e+88) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.75e-87) {
		tmp = -log(x) / n;
	} else if (x <= 1.55e-50) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if (x <= 0.98) {
		tmp = (x - log(x)) / n;
	} else if (x <= 5.1e+88) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.75d-87) then
        tmp = -log(x) / n
    else if (x <= 1.55d-50) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if (x <= 0.98d0) then
        tmp = (x - log(x)) / n
    else if (x <= 5.1d+88) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.75e-87) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.55e-50) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if (x <= 0.98) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 5.1e+88) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.75e-87:
		tmp = -math.log(x) / n
	elif x <= 1.55e-50:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif x <= 0.98:
		tmp = (x - math.log(x)) / n
	elif x <= 5.1e+88:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.75e-87)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.55e-50)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (x <= 0.98)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 5.1e+88)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.75e-87)
		tmp = -log(x) / n;
	elseif (x <= 1.55e-50)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif (x <= 0.98)
		tmp = (x - log(x)) / n;
	elseif (x <= 5.1e+88)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.75e-87], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.55e-50], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.98], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 5.1e+88], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.75000000000000006e-87

   1. Initial program 36.9%

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

   2. Taylor expanded in x around 0 36.8%

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

      1. *-rgt-identity 36.8%

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

      2. associate-*r/ 36.8%

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

      3. unpow-1 36.8%

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

      4. exp-to-pow 36.9%

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

      5. unpow-1 36.9%

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

   4. Simplified 36.9%

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

   5. Taylor expanded in n around inf 56.9%

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

      1. neg-mul-1 56.9%

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

      2. distribute-neg-frac 56.9%

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

   7. Simplified 56.9%

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

   if 1.75000000000000006e-87 < x < 1.5500000000000001e-50

   1. Initial program 67.8%

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

   2. Taylor expanded in n around inf 14.0%

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

      1. log1p-def 14.0%

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

   4. Simplified 14.0%

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

   5. Taylor expanded in x around inf 83.9%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate--l+ 83.9%

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

      2. associate-*r/ 83.9%

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

      3. metadata-eval 83.9%

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

      4. associate-*r/ 83.9%

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

      5. metadata-eval 83.9%

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

      6. unpow2 83.9%

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

   7. Simplified 83.9%

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

   8. Taylor expanded in x around 0 83.9%

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

      1. *-commutative 83.9%

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

   10. Simplified 83.9%

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

   if 1.5500000000000001e-50 < x < 0.97999999999999998

   1. Initial program 43.2%

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

   2. Taylor expanded in n around inf 55.7%

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

      1. log1p-def 55.7%

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

   4. Simplified 55.7%

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

   5. Taylor expanded in x around 0 55.2%

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

      1. neg-mul-1 55.2%

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

      2. unsub-neg 55.2%

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

   7. Simplified 55.2%

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

   if 0.97999999999999998 < x < 5.0999999999999997e88

   1. Initial program 40.2%

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

   2. Taylor expanded in n around inf 40.2%

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

      1. log1p-def 40.2%

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

   4. Simplified 40.2%

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

   5. Taylor expanded in x around inf 67.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate-*r/ 67.2%

         \[\leadsto \frac{\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{n} \]

      2. metadata-eval 67.2%

         \[\leadsto \frac{\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}}{n} \]

      3. unpow2 67.2%

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

   7. Simplified 67.2%

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

   if 5.0999999999999997e88 < x

   1. Initial program 76.4%

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

   2. Taylor expanded in n around inf 76.5%

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

      1. log1p-def 76.5%

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

   4. Simplified 76.5%

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

      1. log1p-udef 76.5%

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

      2. +-commutative 76.5%

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

      3. diff-log 76.5%

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

   6. Applied egg-rr 76.5%

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

   7. Taylor expanded in x around inf 76.5%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{-87}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-50}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 15: 60.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.96:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.96)
   (/ (- x (log x)) n)
   (if (<= x 6.4e+88) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.96) {
		tmp = (x - log(x)) / n;
	} else if (x <= 6.4e+88) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.96d0) then
        tmp = (x - log(x)) / n
    else if (x <= 6.4d+88) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.96) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 6.4e+88) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.96:
		tmp = (x - math.log(x)) / n
	elif x <= 6.4e+88:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.96)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 6.4e+88)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.96)
		tmp = (x - log(x)) / n;
	elseif (x <= 6.4e+88)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.96], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 6.4e+88], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.95999999999999996

   1. Initial program 40.0%

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

   2. Taylor expanded in n around inf 53.3%

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

      1. log1p-def 53.3%

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

   4. Simplified 53.3%

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

   5. Taylor expanded in x around 0 53.2%

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

      1. neg-mul-1 53.2%

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

      2. unsub-neg 53.2%

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

   7. Simplified 53.2%

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

   if 0.95999999999999996 < x < 6.3999999999999997e88

   1. Initial program 40.2%

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

   2. Taylor expanded in n around inf 40.2%

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

      1. log1p-def 40.2%

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

   4. Simplified 40.2%

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

   5. Taylor expanded in x around inf 67.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate-*r/ 67.2%

         \[\leadsto \frac{\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{n} \]

      2. metadata-eval 67.2%

         \[\leadsto \frac{\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}}{n} \]

      3. unpow2 67.2%

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

   7. Simplified 67.2%

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

   if 6.3999999999999997e88 < x

   1. Initial program 76.4%

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

   2. Taylor expanded in n around inf 76.5%

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

      1. log1p-def 76.5%

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

   4. Simplified 76.5%

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

      1. log1p-udef 76.5%

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

      2. +-commutative 76.5%

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

      3. diff-log 76.5%

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

   6. Applied egg-rr 76.5%

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

   7. Taylor expanded in x around inf 76.5%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.96:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 16: 59.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.68)
   (/ (- (log x)) n)
   (if (<= x 6e+88) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.68) {
		tmp = -log(x) / n;
	} else if (x <= 6e+88) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.68d0) then
        tmp = -log(x) / n
    else if (x <= 6d+88) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.68) {
		tmp = -Math.log(x) / n;
	} else if (x <= 6e+88) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.68:
		tmp = -math.log(x) / n
	elif x <= 6e+88:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 6e+88)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.68)
		tmp = -log(x) / n;
	elseif (x <= 6e+88)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.68], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 6e+88], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.680000000000000049

   1. Initial program 40.0%

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

   2. Taylor expanded in x around 0 38.7%

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

      1. *-rgt-identity 38.7%

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

      2. associate-*r/ 38.7%

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

      3. unpow-1 38.7%

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

      4. exp-to-pow 38.7%

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

      5. unpow-1 38.7%

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

   4. Simplified 38.7%

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

   5. Taylor expanded in n around inf 52.9%

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

      1. neg-mul-1 52.9%

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

      2. distribute-neg-frac 52.9%

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

   7. Simplified 52.9%

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

   if 0.680000000000000049 < x < 6.00000000000000011e88

   1. Initial program 40.2%

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

   2. Taylor expanded in n around inf 40.2%

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

      1. log1p-def 40.2%

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

   4. Simplified 40.2%

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

   5. Taylor expanded in x around inf 67.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate-*r/ 67.2%

         \[\leadsto \frac{\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{n} \]

      2. metadata-eval 67.2%

         \[\leadsto \frac{\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}}{n} \]

      3. unpow2 67.2%

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

   7. Simplified 67.2%

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

   if 6.00000000000000011e88 < x

   1. Initial program 76.4%

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

   2. Taylor expanded in n around inf 76.5%

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

      1. log1p-def 76.5%

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

   4. Simplified 76.5%

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

      1. log1p-udef 76.5%

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

      2. +-commutative 76.5%

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

      3. diff-log 76.5%

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

   6. Applied egg-rr 76.5%

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

   7. Taylor expanded in x around inf 76.5%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 17: 47.1% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -10000000.0) (/ 0.0 n) (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -10000000.0) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-10000000.0d0)) then
        tmp = 0.0d0 / n
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -10000000.0) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -10000000.0:
		tmp = 0.0 / n
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -10000000.0)
		tmp = Float64(0.0 / n);
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -10000000.0)
		tmp = 0.0 / n;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -10000000.0], N[(0.0 / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 n) < -1e7

   1. Initial program 100.0%

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

   2. Taylor expanded in n around inf 53.2%

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

      1. log1p-def 53.2%

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

   4. Simplified 53.2%

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

      1. log1p-udef 53.2%

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

      2. +-commutative 53.2%

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

      3. diff-log 53.2%

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

   6. Applied egg-rr 53.2%

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

   7. Taylor expanded in x around inf 54.6%

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

   if -1e7 < (/.f64 1 n)

   1. Initial program 34.5%

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

   2. Taylor expanded in n around inf 60.7%

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

      1. log1p-def 60.7%

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

   4. Simplified 60.7%

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

      1. log1p-udef 60.7%

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

      2. +-commutative 60.7%

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

      3. diff-log 60.9%

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

   6. Applied egg-rr 60.9%

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

   7. Taylor expanded in x around inf 43.0%

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

      1. associate-/r* 44.7%

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

   9. Simplified 44.7%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 18: 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 51.1%

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

2. Taylor expanded in n around inf 58.8%

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

   1. log1p-def 58.8%

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

4. Simplified 58.8%

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

5. Taylor expanded in x around inf 36.9%

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

   1. *-commutative 36.9%

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

7. Simplified 36.9%

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

8. Final simplification 36.9%

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

Alternative 19: 41.1% 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 51.1%

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

2. Taylor expanded in n around inf 58.8%

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

   1. log1p-def 58.8%

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

4. Simplified 58.8%

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

   1. log1p-udef 58.8%

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

   2. +-commutative 58.8%

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

   3. diff-log 58.9%

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

6. Applied egg-rr 58.9%

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

7. Taylor expanded in x around inf 36.9%

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

   1. associate-/r* 38.2%

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

9. Simplified 38.2%

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

10. Final simplification 38.2%

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

Reproduce

herbie shell --seed 2023199 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))