2nthrt (problem 3.4.6)

Percentage Accurate: 52.9% → 83.8%

Time: 42.8s

Alternatives: 16

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: 52.9% 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: 83.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{log1p}\left(x\right)\right)}^{2}\\ t_1 := {\log x}^{2}\\ t_2 := {x}^{\left(\frac{1}{n}\right)}\\ t_3 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ t_4 := \mathsf{fma}\left(0.5, t_0 \cdot {n}^{-2}, t_3\right)\\ t_5 := \mathsf{fma}\left(0.5, t_1 \cdot {n}^{-2}, \frac{\log x}{n}\right)\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\frac{t_2}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-165}:\\ \;\;\;\;\frac{0.5 \cdot \left(t_0 - t_1\right)}{n \cdot n} + \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\frac{1}{\frac{t_4 + t_5}{{t_4}^{2} - {t_5}^{2}}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;{n}^{-1} \cdot \frac{{x}^{\left({n}^{-1}\right)}}{x} + \left(\frac{t_2}{x \cdot x} \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.3333333333333333}{n}}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{t_3} - t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow (log1p x) 2.0))
        (t_1 (pow (log x) 2.0))
        (t_2 (pow x (/ 1.0 n)))
        (t_3 (/ (log1p x) n))
        (t_4 (fma 0.5 (* t_0 (pow n -2.0)) t_3))
        (t_5 (fma 0.5 (* t_1 (pow n -2.0)) (/ (log x) n))))
   (if (<= (/ 1.0 n) -1e-105)
     (/ t_2 (* n x))
     (if (<= (/ 1.0 n) 1e-165)
       (+ (/ (* 0.5 (- t_0 t_1)) (* n n)) (/ (- (log1p x) (log x)) n))
       (if (<= (/ 1.0 n) 2e-139)
         (/ 1.0 (* n x))
         (if (<= (/ 1.0 n) 2e-96)
           (/ 1.0 (/ (+ t_4 t_5) (- (pow t_4 2.0) (pow t_5 2.0))))
           (if (<= (/ 1.0 n) 2e-8)
             (+
              (* (pow n -1.0) (/ (pow x (pow n -1.0)) x))
              (+
               (* (/ t_2 (* x x)) (+ (/ 0.5 (* n n)) (/ -0.5 n)))
               (/ (/ 0.3333333333333333 n) (pow x 3.0))))
             (- (exp t_3) t_2))))))))

C

double code(double x, double n) {
	double t_0 = pow(log1p(x), 2.0);
	double t_1 = pow(log(x), 2.0);
	double t_2 = pow(x, (1.0 / n));
	double t_3 = log1p(x) / n;
	double t_4 = fma(0.5, (t_0 * pow(n, -2.0)), t_3);
	double t_5 = fma(0.5, (t_1 * pow(n, -2.0)), (log(x) / n));
	double tmp;
	if ((1.0 / n) <= -1e-105) {
		tmp = t_2 / (n * x);
	} else if ((1.0 / n) <= 1e-165) {
		tmp = ((0.5 * (t_0 - t_1)) / (n * n)) + ((log1p(x) - log(x)) / n);
	} else if ((1.0 / n) <= 2e-139) {
		tmp = 1.0 / (n * x);
	} else if ((1.0 / n) <= 2e-96) {
		tmp = 1.0 / ((t_4 + t_5) / (pow(t_4, 2.0) - pow(t_5, 2.0)));
	} else if ((1.0 / n) <= 2e-8) {
		tmp = (pow(n, -1.0) * (pow(x, pow(n, -1.0)) / x)) + (((t_2 / (x * x)) * ((0.5 / (n * n)) + (-0.5 / n))) + ((0.3333333333333333 / n) / pow(x, 3.0)));
	} else {
		tmp = exp(t_3) - t_2;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = log1p(x) ^ 2.0
	t_1 = log(x) ^ 2.0
	t_2 = x ^ Float64(1.0 / n)
	t_3 = Float64(log1p(x) / n)
	t_4 = fma(0.5, Float64(t_0 * (n ^ -2.0)), t_3)
	t_5 = fma(0.5, Float64(t_1 * (n ^ -2.0)), Float64(log(x) / n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-105)
		tmp = Float64(t_2 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-165)
		tmp = Float64(Float64(Float64(0.5 * Float64(t_0 - t_1)) / Float64(n * n)) + Float64(Float64(log1p(x) - log(x)) / n));
	elseif (Float64(1.0 / n) <= 2e-139)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-96)
		tmp = Float64(1.0 / Float64(Float64(t_4 + t_5) / Float64((t_4 ^ 2.0) - (t_5 ^ 2.0))));
	elseif (Float64(1.0 / n) <= 2e-8)
		tmp = Float64(Float64((n ^ -1.0) * Float64((x ^ (n ^ -1.0)) / x)) + Float64(Float64(Float64(t_2 / Float64(x * x)) * Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n))) + Float64(Float64(0.3333333333333333 / n) / (x ^ 3.0))));
	else
		tmp = Float64(exp(t_3) - t_2);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 * N[(t$95$0 * N[Power[n, -2.0], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(0.5 * N[(t$95$1 * N[Power[n, -2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-105], N[(t$95$2 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-165], N[(N[(N[(0.5 * N[(t$95$0 - t$95$1), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-139], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-96], N[(1.0 / N[(N[(t$95$4 + t$95$5), $MachinePrecision] / N[(N[Power[t$95$4, 2.0], $MachinePrecision] - N[Power[t$95$5, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-8], N[(N[(N[Power[n, -1.0], $MachinePrecision] * N[(N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$2 / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 / n), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[t$95$3], $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -9.99999999999999965e-106

   1. Initial program 81.4%

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

   2. Taylor expanded in x around inf 93.7%

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

      1. log-rec 93.7%

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

      2. mul-1-neg 93.7%

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

      3. associate-*r/ 93.7%

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

      4. neg-mul-1 93.7%

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

      5. mul-1-neg 93.7%

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

      6. remove-double-neg 93.7%

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

      7. *-lft-identity 93.7%

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

      8. associate-*l/ 93.7%

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

      9. log-pow 93.7%

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

      10. rem-exp-log 93.7%

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

      11. *-commutative 93.7%

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

   4. Simplified 93.7%

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

   if -9.99999999999999965e-106 < (/.f64 1 n) < 1e-165

   1. Initial program 36.6%

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

   2. Taylor expanded in n around 0 36.6%

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

      1. log1p-def 36.6%

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

      2. *-lft-identity 36.6%

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

      3. associate-*l/ 36.6%

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

      4. log-pow 36.6%

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

      5. rem-exp-log 36.6%

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

   4. Simplified 36.6%

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

   5. Taylor expanded in n around -inf 85.0%

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

      1. associate--l+ 85.0%

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

   7. Simplified 85.0%

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

   if 1e-165 < (/.f64 1 n) < 2.00000000000000006e-139

   1. Initial program 30.3%

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

   2. Taylor expanded in x around inf 99.6%

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

      1. log-rec 99.6%

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

      2. mul-1-neg 99.6%

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

      3. associate-*r/ 99.6%

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

      4. neg-mul-1 99.6%

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

      5. mul-1-neg 99.6%

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

      6. remove-double-neg 99.6%

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

      7. *-lft-identity 99.6%

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

      8. associate-*l/ 99.6%

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

      9. log-pow 99.6%

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

      10. rem-exp-log 99.6%

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

      11. *-commutative 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in n around inf 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   if 2.00000000000000006e-139 < (/.f64 1 n) < 1.9999999999999998e-96

   1. Initial program 24.9%

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

   2. Taylor expanded in n around inf 86.2%

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

      1. fma-def 86.2%

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

      2. log1p-def 86.2%

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

      3. unpow2 86.2%

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

      4. log1p-def 86.2%

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

      5. fma-def 86.2%

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

      6. unpow2 86.2%

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

   4. Simplified 86.2%

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

      1. flip-- 86.3%

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

      2. clear-num 86.4%

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

   6. Applied egg-rr 86.4%

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

   if 1.9999999999999998e-96 < (/.f64 1 n) < 2e-8

   1. Initial program 17.5%

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

   2. Taylor expanded in x around inf 79.9%

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

   4. Simplified 79.9%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot x} \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.3333333333333333}{n} + \left(\frac{0.16666666666666666}{{n}^{3}} - \frac{0.5}{n \cdot n}\right)}{\frac{{x}^{3}}{{x}^{\left(\frac{1}{n}\right)}}}\right)} \]
   5. Step-by-step derivation

      1. associate-/r* 80.1%

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

      2. div-inv 80.0%

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

      3. inv-pow 80.0%

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

      4. inv-pow 80.0%

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

   6. Applied egg-rr 80.0%

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

   7. Taylor expanded in n around inf 80.0%

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

      1. associate-/r* 80.0%

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

   9. Simplified 80.0%

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

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

   1. Initial program 57.0%

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

   2. Taylor expanded in n around 0 57.0%

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

      1. log1p-def 93.3%

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

      2. *-lft-identity 93.3%

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

      3. associate-*l/ 93.3%

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

      4. log-pow 93.3%

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

      5. rem-exp-log 93.3%

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

   4. Simplified 93.3%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-165}:\\ \;\;\;\;\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n \cdot n} + \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} \cdot {n}^{-2}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) + \mathsf{fma}\left(0.5, {\log x}^{2} \cdot {n}^{-2}, \frac{\log x}{n}\right)}{{\left(\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} \cdot {n}^{-2}, \frac{\mathsf{log1p}\left(x\right)}{n}\right)\right)}^{2} - {\left(\mathsf{fma}\left(0.5, {\log x}^{2} \cdot {n}^{-2}, \frac{\log x}{n}\right)\right)}^{2}}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;{n}^{-1} \cdot \frac{{x}^{\left({n}^{-1}\right)}}{x} + \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot x} \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.3333333333333333}{n}}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2: 83.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{log1p}\left(x\right)\right)}^{2}\\ t_1 := {\log x}^{2}\\ t_2 := {x}^{\left(\frac{1}{n}\right)}\\ t_3 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ t_4 := \mathsf{fma}\left(0.5 \cdot t_0, {n}^{-2}, t_3\right)\\ t_5 := \mathsf{fma}\left(0.5, t_1 \cdot {n}^{-2}, \frac{\log x}{n}\right)\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\frac{t_2}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-165}:\\ \;\;\;\;\frac{0.5 \cdot \left(t_0 - t_1\right)}{n \cdot n} + \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\frac{{t_4}^{2} - {t_5}^{2}}{t_5 + t_4}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;{n}^{-1} \cdot \frac{{x}^{\left({n}^{-1}\right)}}{x} + \left(\frac{t_2}{x \cdot x} \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.3333333333333333}{n}}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{t_3} - t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow (log1p x) 2.0))
        (t_1 (pow (log x) 2.0))
        (t_2 (pow x (/ 1.0 n)))
        (t_3 (/ (log1p x) n))
        (t_4 (fma (* 0.5 t_0) (pow n -2.0) t_3))
        (t_5 (fma 0.5 (* t_1 (pow n -2.0)) (/ (log x) n))))
   (if (<= (/ 1.0 n) -1e-105)
     (/ t_2 (* n x))
     (if (<= (/ 1.0 n) 1e-165)
       (+ (/ (* 0.5 (- t_0 t_1)) (* n n)) (/ (- (log1p x) (log x)) n))
       (if (<= (/ 1.0 n) 2e-139)
         (/ 1.0 (* n x))
         (if (<= (/ 1.0 n) 2e-96)
           (/ (- (pow t_4 2.0) (pow t_5 2.0)) (+ t_5 t_4))
           (if (<= (/ 1.0 n) 2e-8)
             (+
              (* (pow n -1.0) (/ (pow x (pow n -1.0)) x))
              (+
               (* (/ t_2 (* x x)) (+ (/ 0.5 (* n n)) (/ -0.5 n)))
               (/ (/ 0.3333333333333333 n) (pow x 3.0))))
             (- (exp t_3) t_2))))))))

C

double code(double x, double n) {
	double t_0 = pow(log1p(x), 2.0);
	double t_1 = pow(log(x), 2.0);
	double t_2 = pow(x, (1.0 / n));
	double t_3 = log1p(x) / n;
	double t_4 = fma((0.5 * t_0), pow(n, -2.0), t_3);
	double t_5 = fma(0.5, (t_1 * pow(n, -2.0)), (log(x) / n));
	double tmp;
	if ((1.0 / n) <= -1e-105) {
		tmp = t_2 / (n * x);
	} else if ((1.0 / n) <= 1e-165) {
		tmp = ((0.5 * (t_0 - t_1)) / (n * n)) + ((log1p(x) - log(x)) / n);
	} else if ((1.0 / n) <= 2e-139) {
		tmp = 1.0 / (n * x);
	} else if ((1.0 / n) <= 2e-96) {
		tmp = (pow(t_4, 2.0) - pow(t_5, 2.0)) / (t_5 + t_4);
	} else if ((1.0 / n) <= 2e-8) {
		tmp = (pow(n, -1.0) * (pow(x, pow(n, -1.0)) / x)) + (((t_2 / (x * x)) * ((0.5 / (n * n)) + (-0.5 / n))) + ((0.3333333333333333 / n) / pow(x, 3.0)));
	} else {
		tmp = exp(t_3) - t_2;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = log1p(x) ^ 2.0
	t_1 = log(x) ^ 2.0
	t_2 = x ^ Float64(1.0 / n)
	t_3 = Float64(log1p(x) / n)
	t_4 = fma(Float64(0.5 * t_0), (n ^ -2.0), t_3)
	t_5 = fma(0.5, Float64(t_1 * (n ^ -2.0)), Float64(log(x) / n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-105)
		tmp = Float64(t_2 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-165)
		tmp = Float64(Float64(Float64(0.5 * Float64(t_0 - t_1)) / Float64(n * n)) + Float64(Float64(log1p(x) - log(x)) / n));
	elseif (Float64(1.0 / n) <= 2e-139)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-96)
		tmp = Float64(Float64((t_4 ^ 2.0) - (t_5 ^ 2.0)) / Float64(t_5 + t_4));
	elseif (Float64(1.0 / n) <= 2e-8)
		tmp = Float64(Float64((n ^ -1.0) * Float64((x ^ (n ^ -1.0)) / x)) + Float64(Float64(Float64(t_2 / Float64(x * x)) * Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n))) + Float64(Float64(0.3333333333333333 / n) / (x ^ 3.0))));
	else
		tmp = Float64(exp(t_3) - t_2);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$4 = N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Power[n, -2.0], $MachinePrecision] + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(0.5 * N[(t$95$1 * N[Power[n, -2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-105], N[(t$95$2 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-165], N[(N[(N[(0.5 * N[(t$95$0 - t$95$1), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-139], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-96], N[(N[(N[Power[t$95$4, 2.0], $MachinePrecision] - N[Power[t$95$5, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$5 + t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-8], N[(N[(N[Power[n, -1.0], $MachinePrecision] * N[(N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$2 / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 / n), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[t$95$3], $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -9.99999999999999965e-106

   1. Initial program 81.4%

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

   2. Taylor expanded in x around inf 93.7%

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

      1. log-rec 93.7%

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

      2. mul-1-neg 93.7%

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

      3. associate-*r/ 93.7%

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

      4. neg-mul-1 93.7%

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

      5. mul-1-neg 93.7%

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

      6. remove-double-neg 93.7%

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

      7. *-lft-identity 93.7%

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

      8. associate-*l/ 93.7%

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

      9. log-pow 93.7%

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

      10. rem-exp-log 93.7%

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

      11. *-commutative 93.7%

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

   4. Simplified 93.7%

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

   if -9.99999999999999965e-106 < (/.f64 1 n) < 1e-165

   1. Initial program 36.6%

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

   2. Taylor expanded in n around 0 36.6%

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

      1. log1p-def 36.6%

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

      2. *-lft-identity 36.6%

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

      3. associate-*l/ 36.6%

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

      4. log-pow 36.6%

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

      5. rem-exp-log 36.6%

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

   4. Simplified 36.6%

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

   5. Taylor expanded in n around -inf 85.0%

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

      1. associate--l+ 85.0%

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

   7. Simplified 85.0%

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

   if 1e-165 < (/.f64 1 n) < 2.00000000000000006e-139

   1. Initial program 30.3%

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

   2. Taylor expanded in x around inf 99.6%

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

      1. log-rec 99.6%

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

      2. mul-1-neg 99.6%

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

      3. associate-*r/ 99.6%

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

      4. neg-mul-1 99.6%

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

      5. mul-1-neg 99.6%

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

      6. remove-double-neg 99.6%

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

      7. *-lft-identity 99.6%

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

      8. associate-*l/ 99.6%

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

      9. log-pow 99.6%

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

      10. rem-exp-log 99.6%

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

      11. *-commutative 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in n around inf 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   if 2.00000000000000006e-139 < (/.f64 1 n) < 1.9999999999999998e-96

   1. Initial program 24.9%

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

   2. Taylor expanded in n around inf 86.2%

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

      1. fma-def 86.2%

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

      2. log1p-def 86.2%

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

      3. unpow2 86.2%

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

      4. log1p-def 86.2%

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

      5. fma-def 86.2%

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

      6. unpow2 86.2%

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

   4. Simplified 86.2%

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

      1. flip-- 86.3%

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

      2. div-sub 86.3%

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

      3. sub-neg 86.3%

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

   6. Applied egg-rr 86.3%

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

      1. unsub-neg 86.3%

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

      2. div-sub 86.3%

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

   8. Simplified 86.3%

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

   if 1.9999999999999998e-96 < (/.f64 1 n) < 2e-8

   1. Initial program 17.5%

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

   2. Taylor expanded in x around inf 79.9%

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

   4. Simplified 79.9%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot x} \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.3333333333333333}{n} + \left(\frac{0.16666666666666666}{{n}^{3}} - \frac{0.5}{n \cdot n}\right)}{\frac{{x}^{3}}{{x}^{\left(\frac{1}{n}\right)}}}\right)} \]
   5. Step-by-step derivation

      1. associate-/r* 80.1%

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

      2. div-inv 80.0%

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

      3. inv-pow 80.0%

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

      4. inv-pow 80.0%

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

   6. Applied egg-rr 80.0%

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

   7. Taylor expanded in n around inf 80.0%

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

      1. associate-/r* 80.0%

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

   9. Simplified 80.0%

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

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

   1. Initial program 57.0%

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

   2. Taylor expanded in n around 0 57.0%

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

      1. log1p-def 93.3%

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

      2. *-lft-identity 93.3%

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

      3. associate-*l/ 93.3%

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

      4. log-pow 93.3%

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

      5. rem-exp-log 93.3%

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

   4. Simplified 93.3%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-165}:\\ \;\;\;\;\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n \cdot n} + \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(0.5 \cdot {\left(\mathsf{log1p}\left(x\right)\right)}^{2}, {n}^{-2}, \frac{\mathsf{log1p}\left(x\right)}{n}\right)\right)}^{2} - {\left(\mathsf{fma}\left(0.5, {\log x}^{2} \cdot {n}^{-2}, \frac{\log x}{n}\right)\right)}^{2}}{\mathsf{fma}\left(0.5, {\log x}^{2} \cdot {n}^{-2}, \frac{\log x}{n}\right) + \mathsf{fma}\left(0.5 \cdot {\left(\mathsf{log1p}\left(x\right)\right)}^{2}, {n}^{-2}, \frac{\mathsf{log1p}\left(x\right)}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;{n}^{-1} \cdot \frac{{x}^{\left({n}^{-1}\right)}}{x} + \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot x} \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.3333333333333333}{n}}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 3: 83.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-165}:\\ \;\;\;\;\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n \cdot n} + t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;{n}^{-1} \cdot \frac{{x}^{\left({n}^{-1}\right)}}{x} + \left(\frac{t_0}{x \cdot x} \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.3333333333333333}{n}}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (- (log1p x) (log x)) n)))
   (if (<= (/ 1.0 n) -1e-105)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 1e-165)
       (+ (/ (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0))) (* n n)) t_1)
       (if (<= (/ 1.0 n) 2e-139)
         (/ 1.0 (* n x))
         (if (<= (/ 1.0 n) 2e-96)
           t_1
           (if (<= (/ 1.0 n) 2e-8)
             (+
              (* (pow n -1.0) (/ (pow x (pow n -1.0)) x))
              (+
               (* (/ t_0 (* x x)) (+ (/ 0.5 (* n n)) (/ -0.5 n)))
               (/ (/ 0.3333333333333333 n) (pow x 3.0))))
             (- (exp (/ (log1p x) n)) t_0))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (log1p(x) - log(x)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-105) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-165) {
		tmp = ((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) / (n * n)) + t_1;
	} else if ((1.0 / n) <= 2e-139) {
		tmp = 1.0 / (n * x);
	} else if ((1.0 / n) <= 2e-96) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-8) {
		tmp = (pow(n, -1.0) * (pow(x, pow(n, -1.0)) / x)) + (((t_0 / (x * x)) * ((0.5 / (n * n)) + (-0.5 / n))) + ((0.3333333333333333 / n) / pow(x, 3.0)));
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (Math.log1p(x) - Math.log(x)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-105) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-165) {
		tmp = ((0.5 * (Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0))) / (n * n)) + t_1;
	} else if ((1.0 / n) <= 2e-139) {
		tmp = 1.0 / (n * x);
	} else if ((1.0 / n) <= 2e-96) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-8) {
		tmp = (Math.pow(n, -1.0) * (Math.pow(x, Math.pow(n, -1.0)) / x)) + (((t_0 / (x * x)) * ((0.5 / (n * n)) + (-0.5 / n))) + ((0.3333333333333333 / n) / Math.pow(x, 3.0)));
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (math.log1p(x) - math.log(x)) / n
	tmp = 0
	if (1.0 / n) <= -1e-105:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 1e-165:
		tmp = ((0.5 * (math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0))) / (n * n)) + t_1
	elif (1.0 / n) <= 2e-139:
		tmp = 1.0 / (n * x)
	elif (1.0 / n) <= 2e-96:
		tmp = t_1
	elif (1.0 / n) <= 2e-8:
		tmp = (math.pow(n, -1.0) * (math.pow(x, math.pow(n, -1.0)) / x)) + (((t_0 / (x * x)) * ((0.5 / (n * n)) + (-0.5 / n))) + ((0.3333333333333333 / n) / math.pow(x, 3.0)))
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(log1p(x) - log(x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-105)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-165)
		tmp = Float64(Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) / Float64(n * n)) + t_1);
	elseif (Float64(1.0 / n) <= 2e-139)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-96)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-8)
		tmp = Float64(Float64((n ^ -1.0) * Float64((x ^ (n ^ -1.0)) / x)) + Float64(Float64(Float64(t_0 / Float64(x * x)) * Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n))) + Float64(Float64(0.3333333333333333 / n) / (x ^ 3.0))));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-105], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-165], N[(N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-139], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-96], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-8], N[(N[(N[Power[n, -1.0], $MachinePrecision] * N[(N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 / n), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -9.99999999999999965e-106

   1. Initial program 81.4%

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

   2. Taylor expanded in x around inf 93.7%

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

      1. log-rec 93.7%

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

      2. mul-1-neg 93.7%

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

      3. associate-*r/ 93.7%

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

      4. neg-mul-1 93.7%

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

      5. mul-1-neg 93.7%

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

      6. remove-double-neg 93.7%

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

      7. *-lft-identity 93.7%

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

      8. associate-*l/ 93.7%

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

      9. log-pow 93.7%

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

      10. rem-exp-log 93.7%

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

      11. *-commutative 93.7%

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

   4. Simplified 93.7%

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

   if -9.99999999999999965e-106 < (/.f64 1 n) < 1e-165

   1. Initial program 36.6%

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

   2. Taylor expanded in n around 0 36.6%

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

      1. log1p-def 36.6%

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

      2. *-lft-identity 36.6%

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

      3. associate-*l/ 36.6%

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

      4. log-pow 36.6%

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

      5. rem-exp-log 36.6%

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

   4. Simplified 36.6%

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

   5. Taylor expanded in n around -inf 85.0%

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

      1. associate--l+ 85.0%

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

   7. Simplified 85.0%

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

   if 1e-165 < (/.f64 1 n) < 2.00000000000000006e-139

   1. Initial program 30.3%

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

   2. Taylor expanded in x around inf 99.6%

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

      1. log-rec 99.6%

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

      2. mul-1-neg 99.6%

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

      3. associate-*r/ 99.6%

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

      4. neg-mul-1 99.6%

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

      5. mul-1-neg 99.6%

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

      6. remove-double-neg 99.6%

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

      7. *-lft-identity 99.6%

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

      8. associate-*l/ 99.6%

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

      9. log-pow 99.6%

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

      10. rem-exp-log 99.6%

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

      11. *-commutative 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in n around inf 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   if 2.00000000000000006e-139 < (/.f64 1 n) < 1.9999999999999998e-96

   1. Initial program 24.9%

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

   2. Taylor expanded in n around inf 86.2%

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

      1. log1p-def 86.2%

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

   4. Simplified 86.2%

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

   if 1.9999999999999998e-96 < (/.f64 1 n) < 2e-8

   1. Initial program 17.5%

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

   2. Taylor expanded in x around inf 79.9%

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

   4. Simplified 79.9%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot x} \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.3333333333333333}{n} + \left(\frac{0.16666666666666666}{{n}^{3}} - \frac{0.5}{n \cdot n}\right)}{\frac{{x}^{3}}{{x}^{\left(\frac{1}{n}\right)}}}\right)} \]
   5. Step-by-step derivation

      1. associate-/r* 80.1%

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

      2. div-inv 80.0%

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

      3. inv-pow 80.0%

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

      4. inv-pow 80.0%

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

   6. Applied egg-rr 80.0%

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

   7. Taylor expanded in n around inf 80.0%

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

      1. associate-/r* 80.0%

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

   9. Simplified 80.0%

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

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

   1. Initial program 57.0%

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

   2. Taylor expanded in n around 0 57.0%

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

      1. log1p-def 93.3%

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

      2. *-lft-identity 93.3%

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

      3. associate-*l/ 93.3%

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

      4. log-pow 93.3%

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

      5. rem-exp-log 93.3%

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

   4. Simplified 93.3%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-165}:\\ \;\;\;\;\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n \cdot n} + \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;{n}^{-1} \cdot \frac{{x}^{\left({n}^{-1}\right)}}{x} + \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot x} \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.3333333333333333}{n}}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 4: 83.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;{n}^{-1} \cdot \frac{{x}^{\left({n}^{-1}\right)}}{x} + \left(\frac{t_0}{x \cdot x} \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.3333333333333333}{n}}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (- (log1p x) (log x)) n)))
   (if (<= (/ 1.0 n) -1e-105)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 1e-165)
       t_1
       (if (<= (/ 1.0 n) 2e-139)
         (/ 1.0 (* n x))
         (if (<= (/ 1.0 n) 2e-96)
           t_1
           (if (<= (/ 1.0 n) 2e-8)
             (+
              (* (pow n -1.0) (/ (pow x (pow n -1.0)) x))
              (+
               (* (/ t_0 (* x x)) (+ (/ 0.5 (* n n)) (/ -0.5 n)))
               (/ (/ 0.3333333333333333 n) (pow x 3.0))))
             (- (exp (/ (log1p x) n)) t_0))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (log1p(x) - log(x)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-105) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-165) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-139) {
		tmp = 1.0 / (n * x);
	} else if ((1.0 / n) <= 2e-96) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-8) {
		tmp = (pow(n, -1.0) * (pow(x, pow(n, -1.0)) / x)) + (((t_0 / (x * x)) * ((0.5 / (n * n)) + (-0.5 / n))) + ((0.3333333333333333 / n) / pow(x, 3.0)));
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (Math.log1p(x) - Math.log(x)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-105) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-165) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-139) {
		tmp = 1.0 / (n * x);
	} else if ((1.0 / n) <= 2e-96) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-8) {
		tmp = (Math.pow(n, -1.0) * (Math.pow(x, Math.pow(n, -1.0)) / x)) + (((t_0 / (x * x)) * ((0.5 / (n * n)) + (-0.5 / n))) + ((0.3333333333333333 / n) / Math.pow(x, 3.0)));
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (math.log1p(x) - math.log(x)) / n
	tmp = 0
	if (1.0 / n) <= -1e-105:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 1e-165:
		tmp = t_1
	elif (1.0 / n) <= 2e-139:
		tmp = 1.0 / (n * x)
	elif (1.0 / n) <= 2e-96:
		tmp = t_1
	elif (1.0 / n) <= 2e-8:
		tmp = (math.pow(n, -1.0) * (math.pow(x, math.pow(n, -1.0)) / x)) + (((t_0 / (x * x)) * ((0.5 / (n * n)) + (-0.5 / n))) + ((0.3333333333333333 / n) / math.pow(x, 3.0)))
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(log1p(x) - log(x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-105)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-165)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-139)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-96)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-8)
		tmp = Float64(Float64((n ^ -1.0) * Float64((x ^ (n ^ -1.0)) / x)) + Float64(Float64(Float64(t_0 / Float64(x * x)) * Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n))) + Float64(Float64(0.3333333333333333 / n) / (x ^ 3.0))));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-105], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-165], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-139], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-96], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-8], N[(N[(N[Power[n, -1.0], $MachinePrecision] * N[(N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 / n), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -9.99999999999999965e-106

   1. Initial program 81.4%

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

   2. Taylor expanded in x around inf 93.7%

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

      1. log-rec 93.7%

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

      2. mul-1-neg 93.7%

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

      3. associate-*r/ 93.7%

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

      4. neg-mul-1 93.7%

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

      5. mul-1-neg 93.7%

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

      6. remove-double-neg 93.7%

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

      7. *-lft-identity 93.7%

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

      8. associate-*l/ 93.7%

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

      9. log-pow 93.7%

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

      10. rem-exp-log 93.7%

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

      11. *-commutative 93.7%

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

   4. Simplified 93.7%

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

   if -9.99999999999999965e-106 < (/.f64 1 n) < 1e-165 or 2.00000000000000006e-139 < (/.f64 1 n) < 1.9999999999999998e-96

   1. Initial program 34.6%

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

   2. Taylor expanded in n around inf 85.2%

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

      1. log1p-def 85.2%

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

   4. Simplified 85.2%

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

   if 1e-165 < (/.f64 1 n) < 2.00000000000000006e-139

   1. Initial program 30.3%

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

   2. Taylor expanded in x around inf 99.6%

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

      1. log-rec 99.6%

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

      2. mul-1-neg 99.6%

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

      3. associate-*r/ 99.6%

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

      4. neg-mul-1 99.6%

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

      5. mul-1-neg 99.6%

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

      6. remove-double-neg 99.6%

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

      7. *-lft-identity 99.6%

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

      8. associate-*l/ 99.6%

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

      9. log-pow 99.6%

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

      10. rem-exp-log 99.6%

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

      11. *-commutative 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in n around inf 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   if 1.9999999999999998e-96 < (/.f64 1 n) < 2e-8

   1. Initial program 17.5%

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

   2. Taylor expanded in x around inf 79.9%

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

   4. Simplified 79.9%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot x} \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.3333333333333333}{n} + \left(\frac{0.16666666666666666}{{n}^{3}} - \frac{0.5}{n \cdot n}\right)}{\frac{{x}^{3}}{{x}^{\left(\frac{1}{n}\right)}}}\right)} \]
   5. Step-by-step derivation

      1. associate-/r* 80.1%

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

      2. div-inv 80.0%

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

      3. inv-pow 80.0%

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

      4. inv-pow 80.0%

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

   6. Applied egg-rr 80.0%

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

   7. Taylor expanded in n around inf 80.0%

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

      1. associate-/r* 80.0%

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

   9. Simplified 80.0%

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

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

   1. Initial program 57.0%

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

   2. Taylor expanded in n around 0 57.0%

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

      1. log1p-def 93.3%

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

      2. *-lft-identity 93.3%

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

      3. associate-*l/ 93.3%

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

      4. log-pow 93.3%

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

      5. rem-exp-log 93.3%

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

   4. Simplified 93.3%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-165}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;{n}^{-1} \cdot \frac{{x}^{\left({n}^{-1}\right)}}{x} + \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot x} \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.3333333333333333}{n}}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 5: 83.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t_0}{n \cdot x}\\ t_2 := \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-165}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-96}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq 50000:\\ \;\;\;\;t_1 + \frac{\frac{0.3333333333333333}{{x}^{3}} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1 (/ t_0 (* n x)))
        (t_2 (/ (- (log1p x) (log x)) n)))
   (if (<= (/ 1.0 n) -1e-105)
     t_1
     (if (<= (/ 1.0 n) 1e-165)
       t_2
       (if (<= (/ 1.0 n) 2e-139)
         (/ 1.0 (* n x))
         (if (<= (/ 1.0 n) 2e-96)
           t_2
           (if (<= (/ 1.0 n) 50000.0)
             (+
              t_1
              (/ (- (/ 0.3333333333333333 (pow x 3.0)) (/ 0.5 (* 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 t_1 = t_0 / (n * x);
	double t_2 = (log1p(x) - log(x)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-105) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-165) {
		tmp = t_2;
	} else if ((1.0 / n) <= 2e-139) {
		tmp = 1.0 / (n * x);
	} else if ((1.0 / n) <= 2e-96) {
		tmp = t_2;
	} else if ((1.0 / n) <= 50000.0) {
		tmp = t_1 + (((0.3333333333333333 / pow(x, 3.0)) - (0.5 / (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 t_1 = t_0 / (n * x);
	double t_2 = (Math.log1p(x) - Math.log(x)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-105) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-165) {
		tmp = t_2;
	} else if ((1.0 / n) <= 2e-139) {
		tmp = 1.0 / (n * x);
	} else if ((1.0 / n) <= 2e-96) {
		tmp = t_2;
	} else if ((1.0 / n) <= 50000.0) {
		tmp = t_1 + (((0.3333333333333333 / Math.pow(x, 3.0)) - (0.5 / (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))
	t_1 = t_0 / (n * x)
	t_2 = (math.log1p(x) - math.log(x)) / n
	tmp = 0
	if (1.0 / n) <= -1e-105:
		tmp = t_1
	elif (1.0 / n) <= 1e-165:
		tmp = t_2
	elif (1.0 / n) <= 2e-139:
		tmp = 1.0 / (n * x)
	elif (1.0 / n) <= 2e-96:
		tmp = t_2
	elif (1.0 / n) <= 50000.0:
		tmp = t_1 + (((0.3333333333333333 / math.pow(x, 3.0)) - (0.5 / (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)
	t_1 = Float64(t_0 / Float64(n * x))
	t_2 = Float64(Float64(log1p(x) - log(x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-105)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e-165)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 2e-139)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-96)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 50000.0)
		tmp = Float64(t_1 + Float64(Float64(Float64(0.3333333333333333 / (x ^ 3.0)) - Float64(0.5 / Float64(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]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-105], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-165], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-139], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-96], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], 50000.0], N[(t$95$1 + N[(N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -9.99999999999999965e-106

   1. Initial program 81.4%

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

   2. Taylor expanded in x around inf 93.7%

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

      1. log-rec 93.7%

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

      2. mul-1-neg 93.7%

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

      3. associate-*r/ 93.7%

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

      4. neg-mul-1 93.7%

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

      5. mul-1-neg 93.7%

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

      6. remove-double-neg 93.7%

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

      7. *-lft-identity 93.7%

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

      8. associate-*l/ 93.7%

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

      9. log-pow 93.7%

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

      10. rem-exp-log 93.7%

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

      11. *-commutative 93.7%

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

   4. Simplified 93.7%

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

   if -9.99999999999999965e-106 < (/.f64 1 n) < 1e-165 or 2.00000000000000006e-139 < (/.f64 1 n) < 1.9999999999999998e-96

   1. Initial program 34.6%

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

   2. Taylor expanded in n around inf 85.2%

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

      1. log1p-def 85.2%

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

   4. Simplified 85.2%

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

   if 1e-165 < (/.f64 1 n) < 2.00000000000000006e-139

   1. Initial program 30.3%

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

   2. Taylor expanded in x around inf 99.6%

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

      1. log-rec 99.6%

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

      2. mul-1-neg 99.6%

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

      3. associate-*r/ 99.6%

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

      4. neg-mul-1 99.6%

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

      5. mul-1-neg 99.6%

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

      6. remove-double-neg 99.6%

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

      7. *-lft-identity 99.6%

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

      8. associate-*l/ 99.6%

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

      9. log-pow 99.6%

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

      10. rem-exp-log 99.6%

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

      11. *-commutative 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in n around inf 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   if 1.9999999999999998e-96 < (/.f64 1 n) < 5e4

   1. Initial program 21.0%

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

   2. Taylor expanded in x around inf 70.3%

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

   4. Simplified 70.3%

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

   5. Taylor expanded in n around inf 75.9%

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

      1. associate-*r/ 75.9%

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

      2. metadata-eval 75.9%

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

      3. associate-*r/ 75.9%

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

      4. metadata-eval 75.9%

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

      5. unpow2 75.9%

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

   7. Simplified 75.9%

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

   if 5e4 < (/.f64 1 n)

   1. Initial program 58.4%

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

   2. Taylor expanded in n around 0 58.4%

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

      1. log1p-def 100.0%

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

      2. *-lft-identity 100.0%

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

      3. associate-*l/ 100.0%

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

      4. log-pow 100.0%

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

      5. rem-exp-log 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-165}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 50000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x} + \frac{\frac{0.3333333333333333}{{x}^{3}} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 6: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ t_1 := \frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ t_2 := \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-165}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-96}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq 50000:\\ \;\;\;\;t_1 + \frac{\frac{0.3333333333333333}{{x}^{3}} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\langle \left( \langle \left( t_0 \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x)))
        (t_1 (/ (pow x (/ 1.0 n)) (* n x)))
        (t_2 (/ (- (log1p x) (log x)) n)))
   (if (<= (/ 1.0 n) -1e-105)
     t_1
     (if (<= (/ 1.0 n) 1e-165)
       t_2
       (if (<= (/ 1.0 n) 2e-139)
         t_0
         (if (<= (/ 1.0 n) 2e-96)
           t_2
           (if (<= (/ 1.0 n) 50000.0)
             (+
              t_1
              (/ (- (/ 0.3333333333333333 (pow x 3.0)) (/ 0.5 (* x x))) n))
             (cast
              (!
               :precision
               binary32
               (cast (! :precision binary64 t_0)))))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double t_1 = pow(x, (1.0 / n)) / (n * x);
	double t_2 = (log1p(x) - log(x)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-105) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-165) {
		tmp = t_2;
	} else if ((1.0 / n) <= 2e-139) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-96) {
		tmp = t_2;
	} else if ((1.0 / n) <= 50000.0) {
		tmp = t_1 + (((0.3333333333333333 / pow(x, 3.0)) - (0.5 / (x * x))) / n);
	} else {
		double tmp_3 = t_0;
		double tmp_2 = (float) tmp_3;
		tmp = (double) tmp_2;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	t_1 = Float64((x ^ Float64(1.0 / n)) / Float64(n * x))
	t_2 = Float64(Float64(log1p(x) - log(x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-105)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e-165)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 2e-139)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-96)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 50000.0)
		tmp = Float64(t_1 + Float64(Float64(Float64(0.3333333333333333 / (x ^ 3.0)) - Float64(0.5 / Float64(x * x))) / n));
	else
		tmp_3 = t_0
		tmp_2 = Float32(tmp_3)
		tmp = Float64(tmp_2);
	end
	return tmp
end

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -9.99999999999999965e-106

   1. Initial program 81.4%

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

   2. Taylor expanded in x around inf 93.7%

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

      1. log-rec 93.7%

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

      2. mul-1-neg 93.7%

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

      3. associate-*r/ 93.7%

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

      4. neg-mul-1 93.7%

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

      5. mul-1-neg 93.7%

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

      6. remove-double-neg 93.7%

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

      7. *-lft-identity 93.7%

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

      8. associate-*l/ 93.7%

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

      9. log-pow 93.7%

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

      10. rem-exp-log 93.7%

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

      11. *-commutative 93.7%

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

   4. Simplified 93.7%

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

   if -9.99999999999999965e-106 < (/.f64 1 n) < 1e-165 or 2.00000000000000006e-139 < (/.f64 1 n) < 1.9999999999999998e-96

   1. Initial program 34.6%

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

   2. Taylor expanded in n around inf 85.2%

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

      1. log1p-def 85.2%

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

   4. Simplified 85.2%

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

   if 1e-165 < (/.f64 1 n) < 2.00000000000000006e-139

   1. Initial program 30.3%

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

   2. Taylor expanded in x around inf 99.6%

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

      1. log-rec 99.6%

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

      2. mul-1-neg 99.6%

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

      3. associate-*r/ 99.6%

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

      4. neg-mul-1 99.6%

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

      5. mul-1-neg 99.6%

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

      6. remove-double-neg 99.6%

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

      7. *-lft-identity 99.6%

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

      8. associate-*l/ 99.6%

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

      9. log-pow 99.6%

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

      10. rem-exp-log 99.6%

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

      11. *-commutative 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in n around inf 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   if 1.9999999999999998e-96 < (/.f64 1 n) < 5e4

   1. Initial program 21.0%

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

   2. Taylor expanded in x around inf 70.3%

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

   4. Simplified 70.3%

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

   5. Taylor expanded in n around inf 75.9%

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

      1. associate-*r/ 75.9%

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

      2. metadata-eval 75.9%

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

      3. associate-*r/ 75.9%

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

      4. metadata-eval 75.9%

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

      5. unpow2 75.9%

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

   7. Simplified 75.9%

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

   if 5e4 < (/.f64 1 n)

   1. Initial program 58.4%

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

   2. Taylor expanded in x around inf 1.1%

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

      1. log-rec 1.1%

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

      2. mul-1-neg 1.1%

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

      3. associate-*r/ 1.1%

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

      4. neg-mul-1 1.1%

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

      5. mul-1-neg 1.1%

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

      6. remove-double-neg 1.1%

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

      7. *-lft-identity 1.1%

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

      8. associate-*l/ 1.1%

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

      9. log-pow 1.1%

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

      10. rem-exp-log 1.1%

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

      11. *-commutative 1.1%

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

   4. Simplified 1.1%

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

   5. Taylor expanded in n around inf 31.8%

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

      1. *-commutative 31.8%

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

   7. Simplified 31.8%

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

      1. rewrite-binary64/binary32-simplify 58.5%

         \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\langle \color{blue}{\left( \color{blue}{\frac{1}{x \cdot n}} \right)_{\text{binary64}}} \rangle_{\text{binary32}}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

   9. Applied rewrite-once 58.5%

      \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\langle \color{blue}{\left( \color{blue}{\frac{1}{x \cdot n}} \right)_{\text{binary64}}} \rangle_{\text{binary32}}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-165}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 50000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x} + \frac{\frac{0.3333333333333333}{{x}^{3}} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\langle \left( \langle \left( \frac{1}{n \cdot x} \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}\\ \end{array} \]

Alternative 7: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ t_1 := \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ t_2 := {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+68}:\\ \;\;\;\;\frac{t_2}{x} \cdot \frac{t_2}{n}\\ \mathbf{else}:\\ \;\;\;\;\langle \left( \langle \left( t_0 \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x)))
        (t_1 (/ (- (log1p x) (log x)) n))
        (t_2 (pow x (/ (/ 1.0 n) 2.0))))
   (if (<= (/ 1.0 n) -1e-105)
     (/ (pow x (/ 1.0 n)) (* n x))
     (if (<= (/ 1.0 n) 1e-165)
       t_1
       (if (<= (/ 1.0 n) 2e-139)
         t_0
         (if (<= (/ 1.0 n) 2e-96)
           t_1
           (if (<= (/ 1.0 n) 5e+68)
             (* (/ t_2 x) (/ t_2 n))
             (cast
              (!
               :precision
               binary32
               (cast (! :precision binary64 t_0)))))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double t_1 = (log1p(x) - log(x)) / n;
	double t_2 = pow(x, ((1.0 / n) / 2.0));
	double tmp;
	if ((1.0 / n) <= -1e-105) {
		tmp = pow(x, (1.0 / n)) / (n * x);
	} else if ((1.0 / n) <= 1e-165) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-139) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-96) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+68) {
		tmp = (t_2 / x) * (t_2 / n);
	} else {
		double tmp_3 = t_0;
		double tmp_2 = (float) tmp_3;
		tmp = (double) tmp_2;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	t_1 = Float64(Float64(log1p(x) - log(x)) / n)
	t_2 = x ^ Float64(Float64(1.0 / n) / 2.0)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-105)
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-165)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-139)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-96)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e+68)
		tmp = Float64(Float64(t_2 / x) * Float64(t_2 / n));
	else
		tmp_3 = t_0
		tmp_2 = Float32(tmp_3)
		tmp = Float64(tmp_2);
	end
	return tmp
end

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -9.99999999999999965e-106

   1. Initial program 81.4%

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

   2. Taylor expanded in x around inf 93.7%

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

      1. log-rec 93.7%

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

      2. mul-1-neg 93.7%

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

      3. associate-*r/ 93.7%

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

      4. neg-mul-1 93.7%

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

      5. mul-1-neg 93.7%

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

      6. remove-double-neg 93.7%

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

      7. *-lft-identity 93.7%

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

      8. associate-*l/ 93.7%

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

      9. log-pow 93.7%

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

      10. rem-exp-log 93.7%

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

      11. *-commutative 93.7%

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

   4. Simplified 93.7%

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

   if -9.99999999999999965e-106 < (/.f64 1 n) < 1e-165 or 2.00000000000000006e-139 < (/.f64 1 n) < 1.9999999999999998e-96

   1. Initial program 34.6%

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

   2. Taylor expanded in n around inf 85.2%

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

      1. log1p-def 85.2%

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

   4. Simplified 85.2%

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

   if 1e-165 < (/.f64 1 n) < 2.00000000000000006e-139

   1. Initial program 30.3%

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

   2. Taylor expanded in x around inf 99.6%

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

      1. log-rec 99.6%

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

      2. mul-1-neg 99.6%

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

      3. associate-*r/ 99.6%

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

      4. neg-mul-1 99.6%

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

      5. mul-1-neg 99.6%

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

      6. remove-double-neg 99.6%

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

      7. *-lft-identity 99.6%

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

      8. associate-*l/ 99.6%

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

      9. log-pow 99.6%

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

      10. rem-exp-log 99.6%

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

      11. *-commutative 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in n around inf 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   if 1.9999999999999998e-96 < (/.f64 1 n) < 5.0000000000000004e68

   1. Initial program 29.6%

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

   2. Taylor expanded in x around inf 67.7%

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

      1. log-rec 67.7%

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

      2. mul-1-neg 67.7%

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

      3. associate-*r/ 67.7%

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

      4. neg-mul-1 67.7%

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

      5. mul-1-neg 67.7%

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

      6. remove-double-neg 67.7%

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

      7. *-lft-identity 67.7%

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

      8. associate-*l/ 67.7%

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

      9. log-pow 67.7%

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

      10. rem-exp-log 67.7%

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

      11. *-commutative 67.7%

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

   4. Simplified 67.7%

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

      1. sqr-pow 67.7%

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

      2. times-frac 67.8%

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

      3. inv-pow 67.8%

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

      4. inv-pow 67.8%

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

   6. Applied egg-rr 67.8%

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

      1. unpow-1 67.8%

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

      2. unpow-1 67.8%

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

   8. Simplified 67.8%

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

   if 5.0000000000000004e68 < (/.f64 1 n)

   1. Initial program 51.5%

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

   2. Taylor expanded in x around inf 0.8%

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

      1. log-rec 0.8%

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

      2. mul-1-neg 0.8%

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

      3. associate-*r/ 0.8%

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

      4. neg-mul-1 0.8%

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

      5. mul-1-neg 0.8%

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

      6. remove-double-neg 0.8%

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

      7. *-lft-identity 0.8%

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

      8. associate-*l/ 0.8%

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

      9. log-pow 0.8%

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

      10. rem-exp-log 0.8%

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

      11. *-commutative 0.8%

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

   4. Simplified 0.8%

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

   5. Taylor expanded in n around inf 36.4%

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

      1. *-commutative 36.4%

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

   7. Simplified 36.4%

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

      1. rewrite-binary64/binary32-simplify 67.7%

         \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\langle \color{blue}{\left( \color{blue}{\frac{1}{x \cdot n}} \right)_{\text{binary64}}} \rangle_{\text{binary32}}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

   9. Applied rewrite-once 67.7%

      \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\langle \color{blue}{\left( \color{blue}{\frac{1}{x \cdot n}} \right)_{\text{binary64}}} \rangle_{\text{binary32}}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-165}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+68}:\\ \;\;\;\;\frac{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}{x} \cdot \frac{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;\langle \left( \langle \left( \frac{1}{n \cdot x} \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}\\ \end{array} \]

Alternative 8: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ t_2 := \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\frac{t_1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-165}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-96}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+48}:\\ \;\;\;\;t_1 \cdot \frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\langle \left( \langle \left( t_0 \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x)))
        (t_1 (pow x (/ 1.0 n)))
        (t_2 (/ (- (log1p x) (log x)) n)))
   (if (<= (/ 1.0 n) -1e-105)
     (/ t_1 (* n x))
     (if (<= (/ 1.0 n) 1e-165)
       t_2
       (if (<= (/ 1.0 n) 2e-139)
         t_0
         (if (<= (/ 1.0 n) 2e-96)
           t_2
           (if (<= (/ 1.0 n) 5e+48)
             (* t_1 (/ (/ 1.0 n) x))
             (cast
              (!
               :precision
               binary32
               (cast (! :precision binary64 t_0)))))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double t_1 = pow(x, (1.0 / n));
	double t_2 = (log1p(x) - log(x)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-105) {
		tmp = t_1 / (n * x);
	} else if ((1.0 / n) <= 1e-165) {
		tmp = t_2;
	} else if ((1.0 / n) <= 2e-139) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-96) {
		tmp = t_2;
	} else if ((1.0 / n) <= 5e+48) {
		tmp = t_1 * ((1.0 / n) / x);
	} else {
		double tmp_3 = t_0;
		double tmp_2 = (float) tmp_3;
		tmp = (double) tmp_2;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	t_1 = x ^ Float64(1.0 / n)
	t_2 = Float64(Float64(log1p(x) - log(x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-105)
		tmp = Float64(t_1 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-165)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 2e-139)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-96)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 5e+48)
		tmp = Float64(t_1 * Float64(Float64(1.0 / n) / x));
	else
		tmp_3 = t_0
		tmp_2 = Float32(tmp_3)
		tmp = Float64(tmp_2);
	end
	return tmp
end

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -9.99999999999999965e-106

   1. Initial program 81.4%

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

   2. Taylor expanded in x around inf 93.7%

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

      1. log-rec 93.7%

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

      2. mul-1-neg 93.7%

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

      3. associate-*r/ 93.7%

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

      4. neg-mul-1 93.7%

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

      5. mul-1-neg 93.7%

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

      6. remove-double-neg 93.7%

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

      7. *-lft-identity 93.7%

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

      8. associate-*l/ 93.7%

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

      9. log-pow 93.7%

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

      10. rem-exp-log 93.7%

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

      11. *-commutative 93.7%

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

   4. Simplified 93.7%

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

   if -9.99999999999999965e-106 < (/.f64 1 n) < 1e-165 or 2.00000000000000006e-139 < (/.f64 1 n) < 1.9999999999999998e-96

   1. Initial program 34.6%

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

   2. Taylor expanded in n around inf 85.2%

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

      1. log1p-def 85.2%

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

   4. Simplified 85.2%

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

   if 1e-165 < (/.f64 1 n) < 2.00000000000000006e-139

   1. Initial program 30.3%

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

   2. Taylor expanded in x around inf 99.6%

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

      1. log-rec 99.6%

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

      2. mul-1-neg 99.6%

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

      3. associate-*r/ 99.6%

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

      4. neg-mul-1 99.6%

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

      5. mul-1-neg 99.6%

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

      6. remove-double-neg 99.6%

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

      7. *-lft-identity 99.6%

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

      8. associate-*l/ 99.6%

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

      9. log-pow 99.6%

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

      10. rem-exp-log 99.6%

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

      11. *-commutative 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in n around inf 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   if 1.9999999999999998e-96 < (/.f64 1 n) < 4.99999999999999973e48

   1. Initial program 27.6%

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

   2. Taylor expanded in x around inf 69.5%

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

      1. log-rec 69.5%

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

      2. mul-1-neg 69.5%

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

      3. associate-*r/ 69.5%

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

      4. neg-mul-1 69.5%

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

      5. mul-1-neg 69.5%

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

      6. remove-double-neg 69.5%

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

      7. *-lft-identity 69.5%

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

      8. associate-*l/ 69.5%

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

      9. log-pow 69.5%

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

      10. rem-exp-log 69.5%

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

      11. *-commutative 69.5%

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

   4. Simplified 69.5%

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

      1. div-inv 69.5%

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

      2. inv-pow 69.5%

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

      3. *-commutative 69.5%

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

      4. associate-/r* 69.6%

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

      5. inv-pow 69.6%

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

   6. Applied egg-rr 69.6%

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

      1. *-commutative 69.6%

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

      2. unpow-1 69.6%

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

      3. unpow-1 69.6%

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

   8. Simplified 69.6%

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

   if 4.99999999999999973e48 < (/.f64 1 n)

   1. Initial program 53.4%

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

   2. Taylor expanded in x around inf 0.9%

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

      1. log-rec 0.9%

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

      2. mul-1-neg 0.9%

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

      3. associate-*r/ 0.9%

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

      4. neg-mul-1 0.9%

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

      5. mul-1-neg 0.9%

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

      6. remove-double-neg 0.9%

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

      7. *-lft-identity 0.9%

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

      8. associate-*l/ 0.9%

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

      9. log-pow 0.9%

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

      10. rem-exp-log 0.9%

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

      11. *-commutative 0.9%

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

   4. Simplified 0.9%

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

   5. Taylor expanded in n around inf 35.1%

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

      1. *-commutative 35.1%

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

   7. Simplified 35.1%

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

      1. rewrite-binary64/binary32-simplify 65.1%

         \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\langle \color{blue}{\left( \color{blue}{\frac{1}{x \cdot n}} \right)_{\text{binary64}}} \rangle_{\text{binary32}}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

   9. Applied rewrite-once 65.1%

      \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\langle \color{blue}{\left( \color{blue}{\frac{1}{x \cdot n}} \right)_{\text{binary64}}} \rangle_{\text{binary32}}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-165}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+48}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\langle \left( \langle \left( \frac{1}{n \cdot x} \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}\\ \end{array} \]

Alternative 9: 69.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log x}{n}\\ t_1 := \langle \left( \langle \left( \frac{1}{n \cdot x} \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}\\ t_2 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 2.35 \cdot 10^{-139}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_2\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log x)) n))
        (t_1
         (cast
          (!
           :precision
           binary32
           (cast (! :precision binary64 (/ 1.0 (* n x)))))))
        (t_2 (pow x (/ 1.0 n))))
   (if (<= x 2.35e-139)
     (- (+ 1.0 (/ x n)) t_2)
     (if (<= x 7.2e-73)
       t_0
       (if (<= x 3.1e-30)
         t_1
         (if (<= x 2.6e-20) t_0 (if (<= x 1.0) t_1 (/ t_2 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = -log(x) / n;
	double tmp_2 = 1.0 / (n * x);
	double tmp_1 = (float) tmp_2;
	double t_1 = (double) tmp_1;
	double t_2 = pow(x, (1.0 / n));
	double tmp_3;
	if (x <= 2.35e-139) {
		tmp_3 = (1.0 + (x / n)) - t_2;
	} else if (x <= 7.2e-73) {
		tmp_3 = t_0;
	} else if (x <= 3.1e-30) {
		tmp_3 = t_1;
	} else if (x <= 2.6e-20) {
		tmp_3 = t_0;
	} else if (x <= 1.0) {
		tmp_3 = t_1;
	} else {
		tmp_3 = t_2 / (n * x);
	}
	return tmp_3;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = -log(x) / n
    tmp_2 = 1.0d0 / (n * x)
    tmp_1 = real(tmp_2, 4)
    t_1 = real(tmp_1, 8)
    t_2 = x ** (1.0d0 / n)
    if (x <= 2.35d-139) then
        tmp_3 = (1.0d0 + (x / n)) - t_2
    else if (x <= 7.2d-73) then
        tmp_3 = t_0
    else if (x <= 3.1d-30) then
        tmp_3 = t_1
    else if (x <= 2.6d-20) then
        tmp_3 = t_0
    else if (x <= 1.0d0) then
        tmp_3 = t_1
    else
        tmp_3 = t_2 / (n * x)
    end if
    code = tmp_3
end function

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(x)) / n)
	tmp_2 = Float64(1.0 / Float64(n * x))
	tmp_1 = Float32(tmp_2)
	t_1 = Float64(tmp_1)
	t_2 = x ^ Float64(1.0 / n)
	tmp_3 = 0.0
	if (x <= 2.35e-139)
		tmp_3 = Float64(Float64(1.0 + Float64(x / n)) - t_2);
	elseif (x <= 7.2e-73)
		tmp_3 = t_0;
	elseif (x <= 3.1e-30)
		tmp_3 = t_1;
	elseif (x <= 2.6e-20)
		tmp_3 = t_0;
	elseif (x <= 1.0)
		tmp_3 = t_1;
	else
		tmp_3 = Float64(t_2 / Float64(n * x));
	end
	return tmp_3
end

MATLAB

function tmp_5 = code(x, n)
	t_0 = -log(x) / n;
	tmp_3 = 1.0 / (n * x);
	tmp_2 = single(tmp_3);
	t_1 = double(tmp_2);
	t_2 = x ^ (1.0 / n);
	tmp_4 = 0.0;
	if (x <= 2.35e-139)
		tmp_4 = (1.0 + (x / n)) - t_2;
	elseif (x <= 7.2e-73)
		tmp_4 = t_0;
	elseif (x <= 3.1e-30)
		tmp_4 = t_1;
	elseif (x <= 2.6e-20)
		tmp_4 = t_0;
	elseif (x <= 1.0)
		tmp_4 = t_1;
	else
		tmp_4 = t_2 / (n * x);
	end
	tmp_5 = tmp_4;
end

Derivation

1. Split input into 4 regimes

2. if x < 2.35000000000000014e-139

   1. Initial program 61.5%

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

   2. Taylor expanded in x around 0 61.5%

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

   if 2.35000000000000014e-139 < x < 7.1999999999999999e-73 or 3.09999999999999991e-30 < x < 2.59999999999999995e-20

   1. Initial program 28.7%

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

   2. Taylor expanded in x around 0 28.7%

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

      1. *-lft-identity 28.7%

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

      2. associate-*l/ 28.8%

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

      3. log-pow 28.8%

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

      4. rem-exp-log 28.7%

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

   4. Simplified 28.7%

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

   5. Taylor expanded in n around inf 69.2%

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

      1. mul-1-neg 69.2%

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

   7. Simplified 69.2%

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

   if 7.1999999999999999e-73 < x < 3.09999999999999991e-30 or 2.59999999999999995e-20 < x < 1

   1. Initial program 53.0%

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

   2. Taylor expanded in x around inf 40.7%

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

      1. log-rec 40.7%

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

      2. mul-1-neg 40.7%

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

      3. associate-*r/ 40.7%

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

      4. neg-mul-1 40.7%

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

      5. mul-1-neg 40.7%

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

      6. remove-double-neg 40.7%

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

      7. *-lft-identity 40.7%

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

      8. associate-*l/ 40.7%

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

      9. log-pow 40.7%

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

      10. rem-exp-log 40.7%

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

      11. *-commutative 40.7%

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

   4. Simplified 40.7%

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

   5. Taylor expanded in n around inf 13.4%

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

      1. *-commutative 13.4%

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

   7. Simplified 13.4%

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

      1. rewrite-binary64/binary32-simplify 66.1%

         \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\langle \color{blue}{\left( \color{blue}{\frac{1}{x \cdot n}} \right)_{\text{binary64}}} \rangle_{\text{binary32}}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

   9. Applied rewrite-once 66.1%

      \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\langle \color{blue}{\left( \color{blue}{\frac{1}{x \cdot n}} \right)_{\text{binary64}}} \rangle_{\text{binary32}}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

   if 1 < x

   1. Initial program 57.9%

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

   2. Taylor expanded in x around inf 99.3%

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

      1. log-rec 99.3%

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

      2. mul-1-neg 99.3%

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

      3. associate-*r/ 99.3%

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

      4. neg-mul-1 99.3%

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

      5. mul-1-neg 99.3%

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

      6. remove-double-neg 99.3%

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

      7. *-lft-identity 99.3%

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

      8. associate-*l/ 99.3%

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

      9. log-pow 99.3%

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

      10. rem-exp-log 99.3%

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

      11. *-commutative 99.3%

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

   4. Simplified 99.3%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.35 \cdot 10^{-139}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-30}:\\ \;\;\;\;\langle \left( \langle \left( \frac{1}{n \cdot x} \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\langle \left( \langle \left( \frac{1}{n \cdot x} \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \end{array} \]

Alternative 10: 68.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log x}{n}\\ t_1 := \frac{1}{n \cdot x}\\ t_2 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 3.1 \cdot 10^{-139}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_2\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-31}:\\ \;\;\;\;\sqrt[3]{t_1 \cdot \frac{t_1}{n \cdot x}}\\ \mathbf{elif}\;x \leq 0.56:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log x)) n)) (t_1 (/ 1.0 (* n x))) (t_2 (pow x (/ 1.0 n))))
   (if (<= x 3.1e-139)
     (- (+ 1.0 (/ x n)) t_2)
     (if (<= x 9e-73)
       t_0
       (if (<= x 2.8e-31)
         (cbrt (* t_1 (/ t_1 (* n x))))
         (if (<= x 0.56) t_0 (/ t_2 (* n x))))))))

C

double code(double x, double n) {
	double t_0 = -log(x) / n;
	double t_1 = 1.0 / (n * x);
	double t_2 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 3.1e-139) {
		tmp = (1.0 + (x / n)) - t_2;
	} else if (x <= 9e-73) {
		tmp = t_0;
	} else if (x <= 2.8e-31) {
		tmp = cbrt((t_1 * (t_1 / (n * x))));
	} else if (x <= 0.56) {
		tmp = t_0;
	} else {
		tmp = t_2 / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = -Math.log(x) / n;
	double t_1 = 1.0 / (n * x);
	double t_2 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 3.1e-139) {
		tmp = (1.0 + (x / n)) - t_2;
	} else if (x <= 9e-73) {
		tmp = t_0;
	} else if (x <= 2.8e-31) {
		tmp = Math.cbrt((t_1 * (t_1 / (n * x))));
	} else if (x <= 0.56) {
		tmp = t_0;
	} else {
		tmp = t_2 / (n * x);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(x)) / n)
	t_1 = Float64(1.0 / Float64(n * x))
	t_2 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 3.1e-139)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_2);
	elseif (x <= 9e-73)
		tmp = t_0;
	elseif (x <= 2.8e-31)
		tmp = cbrt(Float64(t_1 * Float64(t_1 / Float64(n * x))));
	elseif (x <= 0.56)
		tmp = t_0;
	else
		tmp = Float64(t_2 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 3.1e-139], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[x, 9e-73], t$95$0, If[LessEqual[x, 2.8e-31], N[Power[N[(t$95$1 * N[(t$95$1 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[x, 0.56], t$95$0, N[(t$95$2 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 3.0999999999999999e-139

   1. Initial program 61.5%

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

   2. Taylor expanded in x around 0 61.5%

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

   if 3.0999999999999999e-139 < x < 9e-73 or 2.7999999999999999e-31 < x < 0.56000000000000005

   1. Initial program 36.4%

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

   2. Taylor expanded in x around 0 27.4%

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

      1. *-lft-identity 27.4%

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

      2. associate-*l/ 27.5%

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

      3. log-pow 27.5%

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

      4. rem-exp-log 27.4%

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

   4. Simplified 27.4%

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

   5. Taylor expanded in n around inf 58.8%

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

      1. mul-1-neg 58.8%

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

   7. Simplified 58.8%

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

   if 9e-73 < x < 2.7999999999999999e-31

   1. Initial program 49.4%

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

   2. Taylor expanded in x around inf 49.9%

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

      1. log-rec 49.9%

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

      2. mul-1-neg 49.9%

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

      3. associate-*r/ 49.9%

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

      4. neg-mul-1 49.9%

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

      5. mul-1-neg 49.9%

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

      6. remove-double-neg 49.9%

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

      7. *-lft-identity 49.9%

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

      8. associate-*l/ 49.9%

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

      9. log-pow 49.9%

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

      10. rem-exp-log 49.9%

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

      11. *-commutative 49.9%

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

   4. Simplified 49.9%

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

   5. Taylor expanded in n around inf 15.7%

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

      1. *-commutative 15.7%

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

   7. Simplified 15.7%

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

      1. add-cbrt-cube_binary64 60.5%

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

   9. Applied rewrite-once 60.5%

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

       1. *-commutative 60.5%

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

       2. associate-*l/ 60.5%

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

       3. *-lft-identity 60.5%

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

   11. Simplified 60.5%

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

   if 0.56000000000000005 < x

   1. Initial program 57.9%

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

   2. Taylor expanded in x around inf 99.3%

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

      1. log-rec 99.3%

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

      2. mul-1-neg 99.3%

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

      3. associate-*r/ 99.3%

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

      4. neg-mul-1 99.3%

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

      5. mul-1-neg 99.3%

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

      6. remove-double-neg 99.3%

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

      7. *-lft-identity 99.3%

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

      8. associate-*l/ 99.3%

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

      9. log-pow 99.3%

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

      10. rem-exp-log 99.3%

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

      11. *-commutative 99.3%

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

   4. Simplified 99.3%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{-139}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-73}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-31}:\\ \;\;\;\;\sqrt[3]{\frac{1}{n \cdot x} \cdot \frac{\frac{1}{n \cdot x}}{n \cdot x}}\\ \mathbf{elif}\;x \leq 0.56:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \end{array} \]

Alternative 11: 68.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 8.5 \cdot 10^{-140}:\\ \;\;\;\;1 - t_0\\ \mathbf{elif}\;x \leq 0.56:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 8.5e-140)
     (- 1.0 t_0)
     (if (<= x 0.56) (/ (- (log x)) n) (/ t_0 (* n x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 8.5e-140) {
		tmp = 1.0 - t_0;
	} else if (x <= 0.56) {
		tmp = -log(x) / n;
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 8.5d-140) then
        tmp = 1.0d0 - t_0
    else if (x <= 0.56d0) then
        tmp = -log(x) / n
    else
        tmp = t_0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 8.5e-140) {
		tmp = 1.0 - t_0;
	} else if (x <= 0.56) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 8.5e-140:
		tmp = 1.0 - t_0
	elif x <= 0.56:
		tmp = -math.log(x) / n
	else:
		tmp = t_0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 8.5e-140)
		tmp = Float64(1.0 - t_0);
	elseif (x <= 0.56)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(t_0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 8.5e-140)
		tmp = 1.0 - t_0;
	elseif (x <= 0.56)
		tmp = -log(x) / n;
	else
		tmp = t_0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 8.5e-140], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[x, 0.56], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 8.49999999999999997e-140

   1. Initial program 61.5%

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

   2. Taylor expanded in x around 0 61.5%

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

      1. *-lft-identity 61.5%

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

      2. associate-*l/ 61.5%

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

      3. log-pow 61.5%

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

      4. rem-exp-log 61.5%

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

   4. Simplified 61.5%

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

   if 8.49999999999999997e-140 < x < 0.56000000000000005

   1. Initial program 40.3%

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

   2. Taylor expanded in x around 0 34.0%

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

      1. *-lft-identity 34.0%

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

      2. associate-*l/ 34.1%

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

      3. log-pow 34.1%

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

      4. rem-exp-log 34.0%

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

   4. Simplified 34.0%

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

   5. Taylor expanded in n around inf 51.5%

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

      1. mul-1-neg 51.5%

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

   7. Simplified 51.5%

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

   if 0.56000000000000005 < x

   1. Initial program 57.9%

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

   2. Taylor expanded in x around inf 99.3%

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

      1. log-rec 99.3%

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

      2. mul-1-neg 99.3%

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

      3. associate-*r/ 99.3%

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

      4. neg-mul-1 99.3%

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

      5. mul-1-neg 99.3%

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

      6. remove-double-neg 99.3%

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

      7. *-lft-identity 99.3%

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

      8. associate-*l/ 99.3%

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

      9. log-pow 99.3%

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

      10. rem-exp-log 99.3%

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

      11. *-commutative 99.3%

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

   4. Simplified 99.3%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-140}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.56:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \end{array} \]

Alternative 12: 69.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 6 \cdot 10^{-139}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{elif}\;x \leq 0.56:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 6e-139)
     (- (+ 1.0 (/ x n)) t_0)
     (if (<= x 0.56) (/ (- (log x)) n) (/ t_0 (* n x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 6e-139) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 0.56) {
		tmp = -log(x) / n;
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 6d-139) then
        tmp = (1.0d0 + (x / n)) - t_0
    else if (x <= 0.56d0) then
        tmp = -log(x) / n
    else
        tmp = t_0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 6e-139) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 0.56) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 6e-139:
		tmp = (1.0 + (x / n)) - t_0
	elif x <= 0.56:
		tmp = -math.log(x) / n
	else:
		tmp = t_0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 6e-139)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	elseif (x <= 0.56)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(t_0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 6e-139)
		tmp = (1.0 + (x / n)) - t_0;
	elseif (x <= 0.56)
		tmp = -log(x) / n;
	else
		tmp = t_0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 6e-139], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 0.56], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.9999999999999998e-139

   1. Initial program 61.5%

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

   2. Taylor expanded in x around 0 61.5%

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

   if 5.9999999999999998e-139 < x < 0.56000000000000005

   1. Initial program 40.3%

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

   2. Taylor expanded in x around 0 34.0%

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

      1. *-lft-identity 34.0%

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

      2. associate-*l/ 34.1%

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

      3. log-pow 34.1%

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

      4. rem-exp-log 34.0%

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

   4. Simplified 34.0%

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

   5. Taylor expanded in n around inf 51.5%

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

      1. mul-1-neg 51.5%

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

   7. Simplified 51.5%

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

   if 0.56000000000000005 < x

   1. Initial program 57.9%

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

   2. Taylor expanded in x around inf 99.3%

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

      1. log-rec 99.3%

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

      2. mul-1-neg 99.3%

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

      3. associate-*r/ 99.3%

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

      4. neg-mul-1 99.3%

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

      5. mul-1-neg 99.3%

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

      6. remove-double-neg 99.3%

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

      7. *-lft-identity 99.3%

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

      8. associate-*l/ 99.3%

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

      9. log-pow 99.3%

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

      10. rem-exp-log 99.3%

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

      11. *-commutative 99.3%

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

   4. Simplified 99.3%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-139}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.56:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \end{array} \]

Alternative 13: 56.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-239}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-9}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+237}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) \cdot \frac{1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.5e-239)
   (/ 1.0 (* n x))
   (if (<= x 1.55e-9)
     (/ (- (log x)) n)
     (if (<= x 2.4e+237)
       (+ (/ (/ 1.0 n) x) (* (+ (/ 0.5 (* n n)) (/ -0.5 n)) (/ 1.0 (* x x))))
       0.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.5e-239) {
		tmp = 1.0 / (n * x);
	} else if (x <= 1.55e-9) {
		tmp = -log(x) / n;
	} else if (x <= 2.4e+237) {
		tmp = ((1.0 / n) / x) + (((0.5 / (n * n)) + (-0.5 / n)) * (1.0 / (x * x)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 4.5d-239) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 1.55d-9) then
        tmp = -log(x) / n
    else if (x <= 2.4d+237) then
        tmp = ((1.0d0 / n) / x) + (((0.5d0 / (n * n)) + ((-0.5d0) / n)) * (1.0d0 / (x * x)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.5e-239) {
		tmp = 1.0 / (n * x);
	} else if (x <= 1.55e-9) {
		tmp = -Math.log(x) / n;
	} else if (x <= 2.4e+237) {
		tmp = ((1.0 / n) / x) + (((0.5 / (n * n)) + (-0.5 / n)) * (1.0 / (x * x)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.5e-239:
		tmp = 1.0 / (n * x)
	elif x <= 1.55e-9:
		tmp = -math.log(x) / n
	elif x <= 2.4e+237:
		tmp = ((1.0 / n) / x) + (((0.5 / (n * n)) + (-0.5 / n)) * (1.0 / (x * x)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.5e-239)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 1.55e-9)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 2.4e+237)
		tmp = Float64(Float64(Float64(1.0 / n) / x) + Float64(Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)) * Float64(1.0 / Float64(x * x))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.5e-239)
		tmp = 1.0 / (n * x);
	elseif (x <= 1.55e-9)
		tmp = -log(x) / n;
	elseif (x <= 2.4e+237)
		tmp = ((1.0 / n) / x) + (((0.5 / (n * n)) + (-0.5 / n)) * (1.0 / (x * x)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.5e-239], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e-9], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 2.4e+237], N[(N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < 4.50000000000000013e-239

   1. Initial program 75.2%

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

   2. Taylor expanded in x around inf 53.3%

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

      1. log-rec 53.3%

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

      2. mul-1-neg 53.3%

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

      3. associate-*r/ 53.3%

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

      4. neg-mul-1 53.3%

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

      5. mul-1-neg 53.3%

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

      6. remove-double-neg 53.3%

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

      7. *-lft-identity 53.3%

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

      8. associate-*l/ 53.3%

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

      9. log-pow 53.3%

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

      10. rem-exp-log 53.3%

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

      11. *-commutative 53.3%

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

   4. Simplified 53.3%

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

   5. Taylor expanded in n around inf 53.8%

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

      1. *-commutative 53.8%

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

   7. Simplified 53.8%

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

   if 4.50000000000000013e-239 < x < 1.55000000000000002e-9

   1. Initial program 45.6%

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

   2. Taylor expanded in x around 0 45.5%

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

      1. *-lft-identity 45.5%

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

      2. associate-*l/ 45.6%

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

      3. log-pow 45.6%

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

      4. rem-exp-log 45.6%

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

   4. Simplified 45.6%

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

   5. Taylor expanded in n around inf 48.4%

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

      1. mul-1-neg 48.4%

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

   7. Simplified 48.4%

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

   if 1.55000000000000002e-9 < x < 2.3999999999999999e237

   1. Initial program 45.5%

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

   2. Taylor expanded in x around inf 81.6%

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

   4. Simplified 81.6%

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

   5. Taylor expanded in n around inf 75.1%

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

      1. unpow2 75.1%

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

   7. Simplified 75.1%

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

   8. Taylor expanded in n around inf 70.5%

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

      1. associate-/r* 68.8%

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

   10. Simplified 70.4%

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

   if 2.3999999999999999e237 < x

   1. Initial program 94.7%

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

   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. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      9. log-pow 100.0%

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

      10. rem-exp-log 100.0%

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

      11. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in n around inf 50.7%

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

      1. associate-/r* 50.7%

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

   7. Simplified 50.7%

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

   9. Applied egg-rr 94.7%

      \[\leadsto \color{blue}{0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-239}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-9}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+237}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) \cdot \frac{1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14: 57.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.15 \cdot 10^{-140}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-9}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+237}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) \cdot \frac{1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.15e-140)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 1.55e-9)
     (/ (- (log x)) n)
     (if (<= x 2.4e+237)
       (+ (/ (/ 1.0 n) x) (* (+ (/ 0.5 (* n n)) (/ -0.5 n)) (/ 1.0 (* x x))))
       0.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.15e-140) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 1.55e-9) {
		tmp = -log(x) / n;
	} else if (x <= 2.4e+237) {
		tmp = ((1.0 / n) / x) + (((0.5 / (n * n)) + (-0.5 / n)) * (1.0 / (x * x)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 4.15d-140) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 1.55d-9) then
        tmp = -log(x) / n
    else if (x <= 2.4d+237) then
        tmp = ((1.0d0 / n) / x) + (((0.5d0 / (n * n)) + ((-0.5d0) / n)) * (1.0d0 / (x * x)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.15e-140) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 1.55e-9) {
		tmp = -Math.log(x) / n;
	} else if (x <= 2.4e+237) {
		tmp = ((1.0 / n) / x) + (((0.5 / (n * n)) + (-0.5 / n)) * (1.0 / (x * x)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.15e-140:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 1.55e-9:
		tmp = -math.log(x) / n
	elif x <= 2.4e+237:
		tmp = ((1.0 / n) / x) + (((0.5 / (n * n)) + (-0.5 / n)) * (1.0 / (x * x)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.15e-140)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 1.55e-9)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 2.4e+237)
		tmp = Float64(Float64(Float64(1.0 / n) / x) + Float64(Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)) * Float64(1.0 / Float64(x * x))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.15e-140)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 1.55e-9)
		tmp = -log(x) / n;
	elseif (x <= 2.4e+237)
		tmp = ((1.0 / n) / x) + (((0.5 / (n * n)) + (-0.5 / n)) * (1.0 / (x * x)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.15e-140], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e-9], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 2.4e+237], N[(N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < 4.1499999999999998e-140

   1. Initial program 61.5%

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

   2. Taylor expanded in x around 0 61.5%

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

      1. *-lft-identity 61.5%

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

      2. associate-*l/ 61.5%

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

      3. log-pow 61.5%

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

      4. rem-exp-log 61.5%

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

   4. Simplified 61.5%

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

   if 4.1499999999999998e-140 < x < 1.55000000000000002e-9

   1. Initial program 38.0%

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

   2. Taylor expanded in x around 0 38.0%

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

      1. *-lft-identity 38.0%

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

      2. associate-*l/ 38.0%

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

      3. log-pow 38.0%

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

      4. rem-exp-log 38.0%

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

   4. Simplified 38.0%

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

   5. Taylor expanded in n around inf 55.4%

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

      1. mul-1-neg 55.4%

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

   7. Simplified 55.4%

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

   if 1.55000000000000002e-9 < x < 2.3999999999999999e237

   1. Initial program 45.5%

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

   2. Taylor expanded in x around inf 81.6%

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

   4. Simplified 81.6%

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

   5. Taylor expanded in n around inf 75.1%

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

      1. unpow2 75.1%

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

   7. Simplified 75.1%

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

   8. Taylor expanded in n around inf 70.5%

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

      1. associate-/r* 68.8%

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

   10. Simplified 70.4%

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

   if 2.3999999999999999e237 < x

   1. Initial program 94.7%

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

   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. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      9. log-pow 100.0%

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

      10. rem-exp-log 100.0%

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

      11. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in n around inf 50.7%

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

      1. associate-/r* 50.7%

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

   7. Simplified 50.7%

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

   9. Applied egg-rr 94.7%

      \[\leadsto \color{blue}{0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.15 \cdot 10^{-140}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-9}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+237}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) \cdot \frac{1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15: 42.7% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.25 \cdot 10^{+237}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (if (<= x 3.25e+237) (/ 1.0 (* n x)) 0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.25e+237) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 3.25d+237) then
        tmp = 1.0d0 / (n * x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.25e+237) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.25e+237:
		tmp = 1.0 / (n * x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.25e+237)
		tmp = Float64(1.0 / Float64(n * x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 3.25e+237)
		tmp = 1.0 / (n * x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.25e+237], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.2499999999999999e237

   1. Initial program 48.6%

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

   2. Taylor expanded in x around inf 62.3%

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

      1. log-rec 62.3%

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

      2. mul-1-neg 62.3%

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

      3. associate-*r/ 62.3%

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

      4. neg-mul-1 62.3%

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

      5. mul-1-neg 62.3%

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

      6. remove-double-neg 62.3%

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

      7. *-lft-identity 62.3%

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

      8. associate-*l/ 62.3%

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

      9. log-pow 62.3%

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

      10. rem-exp-log 62.3%

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

      11. *-commutative 62.3%

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

   4. Simplified 62.3%

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

   5. Taylor expanded in n around inf 46.3%

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

      1. *-commutative 46.3%

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

   7. Simplified 46.3%

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

   if 3.2499999999999999e237 < x

   1. Initial program 94.7%

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

   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. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      9. log-pow 100.0%

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

      10. rem-exp-log 100.0%

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

      11. *-commutative 100.0%

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

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   5. Taylor expanded in n around inf 50.7%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   6. Step-by-step derivation

      1. associate-/r* 50.7%

         \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

   7. Simplified 50.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

   9. Applied egg-rr 94.7%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.25 \cdot 10^{+237}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 16: 30.5% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x n) :precision binary64 0.0)

C

double code(double x, double n) {
	return 0.0;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double x, double n) {
	return 0.0;
}

Python

def code(x, n):
	return 0.0

Julia

function code(x, n)
	return 0.0
end

MATLAB

function tmp = code(x, n)
	tmp = 0.0;
end

Wolfram

code[x_, n_] := 0.0

Derivation

1. Initial program 54.5%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

2. Taylor expanded in x around inf 67.2%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
3. Step-by-step derivation

   1. log-rec 67.2%

      \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   2. mul-1-neg 67.2%

      \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

   3. associate-*r/ 67.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}}}{n \cdot x} \]

   4. neg-mul-1 67.2%

      \[\leadsto \frac{e^{\frac{\color{blue}{--1 \cdot \log x}}{n}}}{n \cdot x} \]

   5. mul-1-neg 67.2%

      \[\leadsto \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]

   6. remove-double-neg 67.2%

      \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

   7. *-lft-identity 67.2%

      \[\leadsto \frac{e^{\frac{\color{blue}{1 \cdot \log x}}{n}}}{n \cdot x} \]

   8. associate-*l/ 67.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{1}{n} \cdot \log x}}}{n \cdot x} \]

   9. log-pow 67.2%

      \[\leadsto \frac{e^{\color{blue}{\log \left({x}^{\left(\frac{1}{n}\right)}\right)}}}{n \cdot x} \]

   10. rem-exp-log 67.2%

       \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

   11. *-commutative 67.2%

       \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

4. Simplified 67.2%

   \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

5. Taylor expanded in n around inf 46.9%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
6. Step-by-step derivation

   1. associate-/r* 46.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

7. Simplified 46.9%

   \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

9. Applied egg-rr 29.6%

   \[\leadsto \color{blue}{0} \]

10. Final simplification 29.6%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))