2nthrt (problem 3.4.6)

Percentage Accurate: 54.0% → 85.9%

Time: 31.3s

Precision: binary64

Cost: 125708

• could not determine a ground truth

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

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(x + 1\right)\\ t_1 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ t_2 := \frac{\log x}{n}\\ t_3 := {\left(e^{2}\right)}^{t_2}\\ t_4 := e^{t_2}\\ \mathbf{if}\;n \leq -2.7 \cdot 10^{+79}:\\ \;\;\;\;t_1 - t_2\\ \mathbf{elif}\;n \leq -3.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}{n}\\ \mathbf{elif}\;n \leq -235000:\\ \;\;\;\;\left(0.5 \cdot \frac{{t_0}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{t_0}^{4}}{{n}^{4}} + \left(\frac{-0.16666666666666666 \cdot {\log x}^{3} - -0.16666666666666666 \cdot {t_0}^{3}}{{n}^{3}} + \frac{t_0 - \log x}{n}\right)\right)\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + 0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}}\right)\\ \mathbf{elif}\;n \leq 126000:\\ \;\;\;\;e^{t_1} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 4.6 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(0.5, e^{t_2 \cdot 2 - t_2} \cdot \frac{\frac{2}{n \cdot n} + \frac{-1}{n}}{x \cdot x}, \frac{\frac{t_3}{n}}{x \cdot t_4}\right) + -0.5 \cdot \frac{\frac{t_3}{n \cdot n}}{\left(x \cdot x\right) \cdot t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]

FPCore

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

↓

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (log (+ x 1.0)))
        (t_1 (/ (log1p x) n))
        (t_2 (/ (log x) n))
        (t_3 (pow (exp 2.0) t_2))
        (t_4 (exp t_2)))
   (if (<= n -2.7e+79)
     (- t_1 t_2)
     (if (<= n -3.5e+61)
       (/
        (+ (/ 0.3333333333333333 (pow x 3.0)) (- (/ 1.0 x) (/ 0.5 (* x x))))
        n)
       (if (<= n -235000.0)
         (-
          (+
           (* 0.5 (/ (pow t_0 2.0) (pow n 2.0)))
           (+
            (* 0.041666666666666664 (/ (pow t_0 4.0) (pow n 4.0)))
            (+
             (/
              (-
               (* -0.16666666666666666 (pow (log x) 3.0))
               (* -0.16666666666666666 (pow t_0 3.0)))
              (pow n 3.0))
             (/ (- t_0 (log x)) n))))
          (+
           (* 0.5 (/ (pow (log x) 2.0) (pow n 2.0)))
           (* 0.041666666666666664 (/ (pow (log x) 4.0) (pow n 4.0)))))
         (if (<= n 126000.0)
           (- (exp t_1) (pow x (/ 1.0 n)))
           (if (<= n 4.6e+40)
             (+
              (fma
               0.5
               (*
                (exp (- (* t_2 2.0) t_2))
                (/ (+ (/ 2.0 (* n n)) (/ -1.0 n)) (* x x)))
               (/ (/ t_3 n) (* x t_4)))
              (* -0.5 (/ (/ t_3 (* n n)) (* (* x x) t_4))))
             (/ (- (log1p x) (log x)) n))))))))

C

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

↓

double code(double x, double n) {
	double t_0 = log((x + 1.0));
	double t_1 = log1p(x) / n;
	double t_2 = log(x) / n;
	double t_3 = pow(exp(2.0), t_2);
	double t_4 = exp(t_2);
	double tmp;
	if (n <= -2.7e+79) {
		tmp = t_1 - t_2;
	} else if (n <= -3.5e+61) {
		tmp = ((0.3333333333333333 / pow(x, 3.0)) + ((1.0 / x) - (0.5 / (x * x)))) / n;
	} else if (n <= -235000.0) {
		tmp = ((0.5 * (pow(t_0, 2.0) / pow(n, 2.0))) + ((0.041666666666666664 * (pow(t_0, 4.0) / pow(n, 4.0))) + ((((-0.16666666666666666 * pow(log(x), 3.0)) - (-0.16666666666666666 * pow(t_0, 3.0))) / pow(n, 3.0)) + ((t_0 - log(x)) / n)))) - ((0.5 * (pow(log(x), 2.0) / pow(n, 2.0))) + (0.041666666666666664 * (pow(log(x), 4.0) / pow(n, 4.0))));
	} else if (n <= 126000.0) {
		tmp = exp(t_1) - pow(x, (1.0 / n));
	} else if (n <= 4.6e+40) {
		tmp = fma(0.5, (exp(((t_2 * 2.0) - t_2)) * (((2.0 / (n * n)) + (-1.0 / n)) / (x * x))), ((t_3 / n) / (x * t_4))) + (-0.5 * ((t_3 / (n * n)) / ((x * x) * t_4)));
	} else {
		tmp = (log1p(x) - log(x)) / n;
	}
	return tmp;
}

Julia

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

↓

function code(x, n)
	t_0 = log(Float64(x + 1.0))
	t_1 = Float64(log1p(x) / n)
	t_2 = Float64(log(x) / n)
	t_3 = exp(2.0) ^ t_2
	t_4 = exp(t_2)
	tmp = 0.0
	if (n <= -2.7e+79)
		tmp = Float64(t_1 - t_2);
	elseif (n <= -3.5e+61)
		tmp = Float64(Float64(Float64(0.3333333333333333 / (x ^ 3.0)) + Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x)))) / n);
	elseif (n <= -235000.0)
		tmp = Float64(Float64(Float64(0.5 * Float64((t_0 ^ 2.0) / (n ^ 2.0))) + Float64(Float64(0.041666666666666664 * Float64((t_0 ^ 4.0) / (n ^ 4.0))) + Float64(Float64(Float64(Float64(-0.16666666666666666 * (log(x) ^ 3.0)) - Float64(-0.16666666666666666 * (t_0 ^ 3.0))) / (n ^ 3.0)) + Float64(Float64(t_0 - log(x)) / n)))) - Float64(Float64(0.5 * Float64((log(x) ^ 2.0) / (n ^ 2.0))) + Float64(0.041666666666666664 * Float64((log(x) ^ 4.0) / (n ^ 4.0)))));
	elseif (n <= 126000.0)
		tmp = Float64(exp(t_1) - (x ^ Float64(1.0 / n)));
	elseif (n <= 4.6e+40)
		tmp = Float64(fma(0.5, Float64(exp(Float64(Float64(t_2 * 2.0) - t_2)) * Float64(Float64(Float64(2.0 / Float64(n * n)) + Float64(-1.0 / n)) / Float64(x * x))), Float64(Float64(t_3 / n) / Float64(x * t_4))) + Float64(-0.5 * Float64(Float64(t_3 / Float64(n * n)) / Float64(Float64(x * x) * t_4))));
	else
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	end
	return tmp
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]

↓

code[x_, n_] := Block[{t$95$0 = N[Log[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Exp[2.0], $MachinePrecision], t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Exp[t$95$2], $MachinePrecision]}, If[LessEqual[n, -2.7e+79], N[(t$95$1 - t$95$2), $MachinePrecision], If[LessEqual[n, -3.5e+61], N[(N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -235000.0], N[(N[(N[(0.5 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.041666666666666664 * N[(N[Power[t$95$0, 4.0], $MachinePrecision] / N[Power[n, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(-0.16666666666666666 * N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] - N[(-0.16666666666666666 * N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[n, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[(N[Power[N[Log[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[n, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 126000.0], N[(N[Exp[t$95$1], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.6e+40], N[(N[(0.5 * N[(N[Exp[N[(N[(t$95$2 * 2.0), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(2.0 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 / n), $MachinePrecision] / N[(x * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[(t$95$3 / N[(n * n), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if n < -2.7e79

   1. Initial program 37.5%

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

   2. Taylor expanded in n around inf 88.4%

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

   3. Simplified 88.4%

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

      [Start] 88.4%	\[ \frac{\log \left(1 + x\right) - \log x}{n} \]
      log1p-def [=>] 88.4%	\[ \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Applied egg-rr 88.5%

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

      [Start] 88.4%	\[ \frac{\mathsf{log1p}\left(x\right) - \log x}{n} \]
      div-sub [=>] 88.5%	\[ \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]

   if -2.7e79 < n < -3.50000000000000018e61

   1. Initial program 22.0%

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

   2. Taylor expanded in n around inf 38.4%

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

   3. Simplified 38.4%

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

      [Start] 38.4%	\[ \frac{\log \left(1 + x\right) - \log x}{n} \]
      log1p-def [=>] 38.4%	\[ \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Taylor expanded in x around inf 85.8%

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

   5. Simplified 85.8%

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

      [Start] 85.8%	\[ \frac{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}{n} \]
      associate--l+ [=>] 85.8%	\[ \frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} \]
      associate-*r/ [=>] 85.8%	\[ \frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]
      metadata-eval [=>] 85.8%	\[ \frac{\frac{\color{blue}{0.3333333333333333}}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]
      associate-*r/ [=>] 85.8%	\[ \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)}{n} \]
      metadata-eval [=>] 85.8%	\[ \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}\right)}{n} \]
      unpow2 [=>] 85.8%	\[ \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{\color{blue}{x \cdot x}}\right)}{n} \]

   if -3.50000000000000018e61 < n < -235000

   1. Initial program 18.9%

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

   2. Taylor expanded in n around -inf 80.9%

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

   if -235000 < n < 126000

   1. Initial program 81.0%

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

   2. Taylor expanded in n around 0 81.0%

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

   3. Simplified 97.9%

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

      [Start] 81.0%	\[ e^{\frac{\log \left(1 + x\right)}{n}} - e^{\frac{\log x}{n}} \]
      log1p-def [=>] 97.9%	\[ e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - e^{\frac{\log x}{n}} \]
      *-rgt-identity [<=] 97.9%	\[ e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]
      associate-*r/ [<=] 97.9%	\[ e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]
      unpow-1 [<=] 97.9%	\[ e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\log x \cdot \color{blue}{{n}^{-1}}} \]
      exp-to-pow [=>] 97.9%	\[ e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - \color{blue}{{x}^{\left({n}^{-1}\right)}} \]
      /-rgt-identity [<=] 97.9%	\[ e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1}}{1}\right)}} \]
      metadata-eval [<=] 97.9%	\[ e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{{n}^{-1}}{\color{blue}{\frac{2}{2}}}\right)} \]
      associate-/l* [<=] 97.9%	\[ e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1} \cdot 2}{2}\right)}} \]
      *-commutative [<=] 97.9%	\[ e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{\color{blue}{2 \cdot {n}^{-1}}}{2}\right)} \]
      *-commutative [=>] 97.9%	\[ e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{\color{blue}{{n}^{-1} \cdot 2}}{2}\right)} \]
      associate-/l* [=>] 97.9%	\[ e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1}}{\frac{2}{2}}\right)}} \]
      metadata-eval [=>] 97.9%	\[ e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{{n}^{-1}}{\color{blue}{1}}\right)} \]
      /-rgt-identity [=>] 97.9%	\[ e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left({n}^{-1}\right)}} \]
      unpow-1 [=>] 97.9%	\[ e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   if 126000 < n < 4.59999999999999987e40

   1. Initial program 26.1%

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

   2. Applied egg-rr 26.7%

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

      [Start] 26.1%	\[ {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
      flip-- [=>] 26.1%	\[ \color{blue}{\frac{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}} \]
      div-inv [=>] 26.1%	\[ \color{blue}{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}} \]
      pow2 [=>] 26.1%	\[ \left(\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{2}} - {x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}} \]
      pow-to-exp [=>] 26.1%	\[ \left({\color{blue}{\left(e^{\log \left(x + 1\right) \cdot \frac{1}{n}}\right)}}^{2} - {x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}} \]
      un-div-inv [=>] 26.1%	\[ \left({\left(e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}}}\right)}^{2} - {x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}} \]
      +-commutative [=>] 26.1%	\[ \left({\left(e^{\frac{\log \color{blue}{\left(1 + x\right)}}{n}}\right)}^{2} - {x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}} \]
      log1p-udef [<=] 26.1%	\[ \left({\left(e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}\right)}^{2} - {x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}} \]
      pow-sqr [=>] 26.7%	\[ \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} - \color{blue}{{x}^{\left(2 \cdot \frac{1}{n}\right)}}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}} \]
      inv-pow [=>] 26.7%	\[ \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} - {x}^{\left(2 \cdot \color{blue}{{n}^{-1}}\right)}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}} \]

   3. Simplified 26.9%

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

      [Start] 26.7%	\[ \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} - {x}^{\left(2 \cdot {n}^{-1}\right)}\right) \cdot \frac{1}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} + {x}^{\left({n}^{-1}\right)}} \]
      associate-*r/ [=>] 26.7%	\[ \color{blue}{\frac{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} - {x}^{\left(2 \cdot {n}^{-1}\right)}\right) \cdot 1}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} + {x}^{\left({n}^{-1}\right)}}} \]

   4. Taylor expanded in x around inf 73.0%

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

   5. Simplified 81.7%

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

      [Start] 73.0%

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

      cancel-sign-sub-inv [=>] 73.0%

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

   if 4.59999999999999987e40 < n

   1. Initial program 19.7%

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

   2. Taylor expanded in n around inf 76.6%

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

   3. Simplified 76.6%

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

      [Start] 76.6%	\[ \frac{\log \left(1 + x\right) - \log x}{n} \]
      log1p-def [=>] 76.6%	\[ \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

3. Recombined 6 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{+79}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;n \leq -3.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}{n}\\ \mathbf{elif}\;n \leq -235000:\\ \;\;\;\;\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{\log \left(x + 1\right)}^{4}}{{n}^{4}} + \left(\frac{-0.16666666666666666 \cdot {\log x}^{3} - -0.16666666666666666 \cdot {\log \left(x + 1\right)}^{3}}{{n}^{3}} + \frac{\log \left(x + 1\right) - \log x}{n}\right)\right)\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + 0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}}\right)\\ \mathbf{elif}\;n \leq 126000:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 4.6 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(0.5, e^{\frac{\log x}{n} \cdot 2 - \frac{\log x}{n}} \cdot \frac{\frac{2}{n \cdot n} + \frac{-1}{n}}{x \cdot x}, \frac{\frac{{\left(e^{2}\right)}^{\left(\frac{\log x}{n}\right)}}{n}}{x \cdot e^{\frac{\log x}{n}}}\right) + -0.5 \cdot \frac{\frac{{\left(e^{2}\right)}^{\left(\frac{\log x}{n}\right)}}{n \cdot n}}{\left(x \cdot x\right) \cdot e^{\frac{\log x}{n}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	85.9%
Cost	125708

\[\begin{array}{l} t_0 := \log \left(x + 1\right)\\ t_1 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ t_2 := \frac{\log x}{n}\\ t_3 := {\left(e^{2}\right)}^{t_2}\\ t_4 := e^{t_2}\\ \mathbf{if}\;n \leq -2.7 \cdot 10^{+79}:\\ \;\;\;\;t_1 - t_2\\ \mathbf{elif}\;n \leq -3.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}{n}\\ \mathbf{elif}\;n \leq -235000:\\ \;\;\;\;\left(0.5 \cdot \frac{{t_0}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{t_0}^{4}}{{n}^{4}} + \left(\frac{-0.16666666666666666 \cdot {\log x}^{3} - -0.16666666666666666 \cdot {t_0}^{3}}{{n}^{3}} + \frac{t_0 - \log x}{n}\right)\right)\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + 0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}}\right)\\ \mathbf{elif}\;n \leq 126000:\\ \;\;\;\;e^{t_1} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 4.6 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(0.5, e^{t_2 \cdot 2 - t_2} \cdot \frac{\frac{2}{n \cdot n} + \frac{-1}{n}}{x \cdot x}, \frac{\frac{t_3}{n}}{x \cdot t_4}\right) + -0.5 \cdot \frac{\frac{t_3}{n \cdot n}}{\left(x \cdot x\right) \cdot t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]

Alternative 2
Accuracy	85.7%
Cost	93780

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ t_1 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ t_2 := \frac{\log x}{n}\\ t_3 := e^{t_2}\\ t_4 := {\left(e^{2}\right)}^{t_2}\\ \mathbf{if}\;n \leq -2.7 \cdot 10^{+79}:\\ \;\;\;\;t_1 - t_2\\ \mathbf{elif}\;n \leq -3.1 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}{n}\\ \mathbf{elif}\;n \leq -1450000:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, t_0\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{{n}^{3}}\right) + -0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\\ \mathbf{elif}\;n \leq 245000:\\ \;\;\;\;\log \left(e^{-{x}^{\left({n}^{-1}\right)}} \cdot e^{e^{t_1}}\right)\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(0.5, e^{t_2 \cdot 2 - t_2} \cdot \frac{\frac{2}{n \cdot n} + \frac{-1}{n}}{x \cdot x}, \frac{\frac{t_4}{n}}{x \cdot t_3}\right) + -0.5 \cdot \frac{\frac{t_4}{n \cdot n}}{\left(x \cdot x\right) \cdot t_3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	85.7%
Cost	79244

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ t_1 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ \mathbf{if}\;n \leq -2.7 \cdot 10^{+81}:\\ \;\;\;\;t_1 - \frac{\log x}{n}\\ \mathbf{elif}\;n \leq -3.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}{n}\\ \mathbf{elif}\;n \leq -1450000:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, t_0\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{{n}^{3}}\right) + -0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\\ \mathbf{elif}\;n \leq 2900000:\\ \;\;\;\;\log \left(e^{-{x}^{\left({n}^{-1}\right)}} \cdot e^{e^{t_1}}\right)\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x} + \frac{\frac{-0.5}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	85.7%
Cost	46668

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ t_1 := \frac{\log x}{n}\\ \mathbf{if}\;n \leq -2.7 \cdot 10^{+79}:\\ \;\;\;\;t_0 - t_1\\ \mathbf{elif}\;n \leq -3.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}{n}\\ \mathbf{elif}\;n \leq -29000000:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, t_0\right) - \left(t_1 + 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right)\\ \mathbf{elif}\;n \leq 1360000:\\ \;\;\;\;\log \left(e^{-{x}^{\left({n}^{-1}\right)}} \cdot e^{e^{t_0}}\right)\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x} + \frac{\frac{-0.5}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]

Alternative 5
Accuracy	85.7%
Cost	46540

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ t_1 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ \mathbf{if}\;n \leq -2.7 \cdot 10^{+79}:\\ \;\;\;\;t_1 - \frac{\log x}{n}\\ \mathbf{elif}\;n \leq -1.8 \cdot 10^{+59}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}{n}\\ \mathbf{elif}\;n \leq -15500000:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, t_0\right) + -0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\\ \mathbf{elif}\;n \leq 2900000:\\ \;\;\;\;\log \left(e^{-{x}^{\left({n}^{-1}\right)}} \cdot e^{e^{t_1}}\right)\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x} + \frac{\frac{-0.5}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	79.6%
Cost	46084

\[\begin{array}{l} \mathbf{if}\;x \leq 0.65:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{{\log x}^{2}}{n \cdot n}, -0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right) - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{n \cdot x}\\ \end{array} \]

Alternative 7
Accuracy	85.6%
Cost	45840

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ t_1 := t_0 - \frac{\log x}{n}\\ \mathbf{if}\;n \leq -2.7 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq -2.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}{n}\\ \mathbf{elif}\;n \leq -12500000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 2900000:\\ \;\;\;\;\log \left(e^{-{x}^{\left({n}^{-1}\right)}} \cdot e^{e^{t_0}}\right)\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x} + \frac{\frac{-0.5}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]

Alternative 8
Accuracy	85.4%
Cost	20364

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ t_1 := \frac{\log x}{n}\\ \mathbf{if}\;n \leq -5 \cdot 10^{+79}:\\ \;\;\;\;t_0 - t_1\\ \mathbf{elif}\;n \leq -3.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}{n}\\ \mathbf{elif}\;n \leq -105000000:\\ \;\;\;\;-0.5 \cdot \frac{{\log x}^{2}}{n \cdot n} - t_1\\ \mathbf{elif}\;n \leq 2800000:\\ \;\;\;\;e^{t_0} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x} + \frac{\frac{-0.5}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]

Alternative 9
Accuracy	85.8%
Cost	20240

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ t_1 := t_0 - \frac{\log x}{n}\\ \mathbf{if}\;n \leq -5.6 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq -3.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}{n}\\ \mathbf{elif}\;n \leq -16000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 2300000:\\ \;\;\;\;e^{t_0} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x} + \frac{\frac{-0.5}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]

Alternative 10
Accuracy	81.8%
Cost	14160

\[\begin{array}{l} t_0 := 1 + \frac{x}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ t_2 := \frac{0.5}{n \cdot n}\\ t_3 := \left(x \cdot x\right) \cdot \left(\frac{0.5}{n} - t_2\right)\\ t_4 := \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -500:\\ \;\;\;\;\frac{t_1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-54}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-41}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x} + \frac{\frac{-0.5}{n}}{x \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+152}:\\ \;\;\;\;\frac{t_0 \cdot t_0 + \left(\left(x \cdot x\right) \cdot \left(t_2 - \frac{0.5}{n}\right)\right) \cdot t_3}{t_0 + t_3} - t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(-\log x\right)}{n \cdot n}\\ \end{array} \]

Alternative 11
Accuracy	81.8%
Cost	14160

\[\begin{array}{l} t_0 := \frac{0.5}{n \cdot n}\\ t_1 := \left(x \cdot x\right) \cdot \left(\frac{0.5}{n} - t_0\right)\\ t_2 := {x}^{\left(\frac{1}{n}\right)}\\ t_3 := 1 + \frac{x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -500:\\ \;\;\;\;\frac{t_2}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-54}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-41}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x} + \frac{\frac{-0.5}{n}}{x \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+152}:\\ \;\;\;\;\frac{t_3 \cdot t_3 + \left(\left(x \cdot x\right) \cdot \left(t_0 - \frac{0.5}{n}\right)\right) \cdot t_1}{t_3 + t_1} - t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(-\log x\right)}{n \cdot n}\\ \end{array} \]

Alternative 12
Accuracy	82.1%
Cost	14160

\[\begin{array}{l} t_0 := 1 + \frac{x}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ t_2 := \frac{0.5}{n \cdot n}\\ t_3 := \left(x \cdot x\right) \cdot \left(\frac{0.5}{n} - t_2\right)\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-54}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-41}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x} + \frac{\frac{-0.5}{n}}{x \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+152}:\\ \;\;\;\;\frac{t_0 \cdot t_0 + \left(\left(x \cdot x\right) \cdot \left(t_2 - \frac{0.5}{n}\right)\right) \cdot t_3}{t_0 + t_3} - t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(-\log x\right)}{n \cdot n}\\ \end{array} \]

Alternative 13
Accuracy	69.5%
Cost	8544

\[\begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ t_1 := \sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)}\\ t_2 := -\log x\\ t_3 := \frac{t_2}{n}\\ t_4 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 3.1 \cdot 10^{-254}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-227}:\\ \;\;\;\;1 - t_4\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-188}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-180}:\\ \;\;\;\;\frac{n \cdot t_2}{n \cdot n}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-165}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-46}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-12}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot t_0\\ \end{array} \]

Alternative 14
Accuracy	71.0%
Cost	7964

\[\begin{array}{l} t_0 := -\log x\\ t_1 := \frac{t_0}{n}\\ t_2 := \frac{n \cdot t_0}{n \cdot n}\\ t_3 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 3.1 \cdot 10^{-254}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-227}:\\ \;\;\;\;1 - t_3\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-188}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{-85}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-37}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \frac{1}{n \cdot x}\\ \end{array} \]

Alternative 15
Accuracy	71.0%
Cost	7836

\[\begin{array}{l} t_0 := -\log x\\ t_1 := \frac{t_0}{n}\\ t_2 := \frac{n \cdot t_0}{n \cdot n}\\ t_3 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 2.7 \cdot 10^{-254}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-227}:\\ \;\;\;\;1 - t_3\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-188}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{-85}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-37}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{n \cdot x}\\ \end{array} \]

Alternative 16
Accuracy	56.9%
Cost	7704

\[\begin{array}{l} t_0 := -\log x\\ t_1 := \frac{t_0}{n}\\ t_2 := \frac{n \cdot t_0}{n \cdot n}\\ \mathbf{if}\;x \leq 2.8 \cdot 10^{-254}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.02 \cdot 10^{-227}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-188}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{-85}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \]

Alternative 17
Accuracy	55.5%
Cost	7316

\[\begin{array}{l} t_0 := \frac{-\log x}{n}\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.6 \cdot 10^{-254}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-224}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \]

Alternative 18
Accuracy	56.2%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \]

Alternative 19
Accuracy	43.7%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 20
Accuracy	41.2%
Cost	320

\[\frac{1}{n \cdot x} \]

Alternative 21
Accuracy	41.8%
Cost	320

\[\frac{\frac{1}{n}}{x} \]

Alternative 22
Accuracy	41.8%
Cost	320

\[\frac{\frac{1}{x}}{n} \]

Alternative 23
Accuracy	4.5%
Cost	192

\[\frac{x}{n} \]

Reproduce

herbie shell --seed 2023165 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))