2nthrt (problem 3.4.6)

Average Accuracy: 48.5% → 88.8%

Time: 28.9s

Precision: binary64

Cost: 85380

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 5.7:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{{n}^{3}}\right) + \left(\frac{{\log x}^{2}}{n \cdot n} \cdot -0.5 - \frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right), \log x\right)}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{{\left(e^{-0.5}\right)}^{\left(-\frac{\log x}{n}\right)}}{x \cdot n}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)} + {x}^{\left(\frac{0.5}{n}\right)}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x n)
 :precision binary64
 (if (<= x 5.7)
   (+
    (fma
     0.5
     (/ (pow (log1p x) 2.0) (* n n))
     (/
      (* 0.16666666666666666 (- (pow (log1p x) 3.0) (pow (log x) 3.0)))
      (pow n 3.0)))
    (-
     (* (/ (pow (log x) 2.0) (* n n)) -0.5)
     (/ (fma -1.0 (log1p x) (log x)) n)))
   (*
    (* 0.5 (/ (pow (exp -0.5) (- (/ (log x) n))) (* x n)))
    (+ (pow (+ x 1.0) (/ 0.5 n)) (pow x (/ 0.5 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 tmp;
	if (x <= 5.7) {
		tmp = fma(0.5, (pow(log1p(x), 2.0) / (n * n)), ((0.16666666666666666 * (pow(log1p(x), 3.0) - pow(log(x), 3.0))) / pow(n, 3.0))) + (((pow(log(x), 2.0) / (n * n)) * -0.5) - (fma(-1.0, log1p(x), log(x)) / n));
	} else {
		tmp = (0.5 * (pow(exp(-0.5), -(log(x) / n)) / (x * n))) * (pow((x + 1.0), (0.5 / n)) + pow(x, (0.5 / 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)
	tmp = 0.0
	if (x <= 5.7)
		tmp = Float64(fma(0.5, Float64((log1p(x) ^ 2.0) / Float64(n * n)), Float64(Float64(0.16666666666666666 * Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))) / (n ^ 3.0))) + Float64(Float64(Float64((log(x) ^ 2.0) / Float64(n * n)) * -0.5) - Float64(fma(-1.0, log1p(x), log(x)) / n)));
	else
		tmp = Float64(Float64(0.5 * Float64((exp(-0.5) ^ Float64(-Float64(log(x) / n))) / Float64(x * n))) * Float64((Float64(x + 1.0) ^ Float64(0.5 / n)) + (x ^ Float64(0.5 / 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_] := If[LessEqual[x, 5.7], N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(N[(0.16666666666666666 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[n, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] - N[(N[(-1.0 * N[Log[1 + x], $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[Power[N[Exp[-0.5], $MachinePrecision], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision])], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(0.5 / n), $MachinePrecision]], $MachinePrecision] + N[Power[x, N[(0.5 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.70000000000000018

   1. Initial program 26.8%

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

   2. Taylor expanded in n around -inf 78.7%

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

   3. Simplified 78.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{{n}^{3}}\right) - \left(\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right), \log x\right)}{n} - \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\right)} \]
      Proof

      [Start] 78.7	\[ \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \left(-1 \cdot \frac{-0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3} - -0.16666666666666666 \cdot {\log x}^{3}}{{n}^{3}} + -1 \cdot \frac{-1 \cdot \log \left(1 + x\right) - -1 \cdot \log x}{n}\right)\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} \]

   if 5.70000000000000018 < x

   1. Initial program 67.0%

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

   2. Applied egg-rr 67.0%

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

      [Start] 67.0	\[ {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
      add-sqr-sqrt [=>] 66.9	\[ \color{blue}{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - {x}^{\left(\frac{1}{n}\right)} \]
      add-sqr-sqrt [=>] 67.0	\[ \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \color{blue}{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}} \]
      difference-of-squares [=>] 67.0	\[ \color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)} \]
      *-commutative [=>] 67.0	\[ \color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)} \]
      sqrt-pow1 [=>] 67.0	\[ \left(\color{blue}{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \]
      associate-/l/ [=>] 67.0	\[ \left({\left(x + 1\right)}^{\color{blue}{\left(\frac{1}{2 \cdot n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \]
      associate-/r* [=>] 67.0	\[ \left({\left(x + 1\right)}^{\color{blue}{\left(\frac{\frac{1}{2}}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \]
      metadata-eval [=>] 67.0	\[ \left({\left(x + 1\right)}^{\left(\frac{\color{blue}{0.5}}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \]
      sqrt-pow1 [=>] 67.0	\[ \left({\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)} - \color{blue}{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \]
      associate-/l/ [=>] 67.0	\[ \left({\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)} - {x}^{\color{blue}{\left(\frac{1}{2 \cdot n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \]
      associate-/r* [=>] 67.0	\[ \left({\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)} - {x}^{\color{blue}{\left(\frac{\frac{1}{2}}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \]
      metadata-eval [=>] 67.0	\[ \left({\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)} - {x}^{\left(\frac{\color{blue}{0.5}}{n}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \]

   3. Taylor expanded in x around inf 97.4%

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

   4. Simplified 97.4%

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

      [Start] 97.4	\[ \left(0.5 \cdot \frac{e^{-0.5 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)} + {x}^{\left(\frac{0.5}{n}\right)}\right) \]
      exp-prod [=>] 97.4	\[ \left(0.5 \cdot \frac{\color{blue}{{\left(e^{-0.5}\right)}^{\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}}}{n \cdot x}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)} + {x}^{\left(\frac{0.5}{n}\right)}\right) \]
      log-rec [=>] 97.4	\[ \left(0.5 \cdot \frac{{\left(e^{-0.5}\right)}^{\left(\frac{\color{blue}{-\log x}}{n}\right)}}{n \cdot x}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)} + {x}^{\left(\frac{0.5}{n}\right)}\right) \]
      *-commutative [=>] 97.4	\[ \left(0.5 \cdot \frac{{\left(e^{-0.5}\right)}^{\left(\frac{-\log x}{n}\right)}}{\color{blue}{x \cdot n}}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)} + {x}^{\left(\frac{0.5}{n}\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.7:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{{n}^{3}}\right) + \left(\frac{{\log x}^{2}}{n \cdot n} \cdot -0.5 - \frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right), \log x\right)}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{{\left(e^{-0.5}\right)}^{\left(-\frac{\log x}{n}\right)}}{x \cdot n}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)} + {x}^{\left(\frac{0.5}{n}\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	88.9%
Cost	33604

\[\begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)} + {x}^{\left(\frac{0.5}{n}\right)}\\ \mathbf{if}\;x \leq 21000000:\\ \;\;\;\;t_0 \cdot \frac{0.5 \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{{\left(e^{-0.5}\right)}^{\left(-\frac{\log x}{n}\right)}}{x \cdot n}\right) \cdot t_0\\ \end{array} \]

Alternative 2
Accuracy	88.9%
Cost	26948

\[\begin{array}{l} \mathbf{if}\;x \leq 21000000:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)} + {x}^{\left(\frac{0.5}{n}\right)}\right) \cdot \frac{0.5 \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]

Alternative 3
Accuracy	88.6%
Cost	20612

\[\begin{array}{l} \mathbf{if}\;x \leq 5.7:\\ \;\;\;\;\frac{0.5 \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n} \cdot \left({x}^{\left(\frac{0.5}{n}\right)} + \left(1 + 0.5 \cdot \frac{x}{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]

Alternative 4
Accuracy	80.6%
Cost	13508

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-55}:\\ \;\;\;\;\frac{0.5 \cdot \frac{1}{x}}{n} \cdot \left(1 + {\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 5
Accuracy	80.6%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-55}:\\ \;\;\;\;\frac{0.5 \cdot \frac{1}{x}}{n} \cdot \left(1 + {\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 6
Accuracy	88.6%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 5.7:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]

Alternative 7
Accuracy	80.5%
Cost	7944

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-55}:\\ \;\;\;\;\frac{0.5 \cdot \frac{1}{x}}{n} \cdot \left(1 + {\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 8
Accuracy	80.5%
Cost	7820

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 9
Accuracy	80.4%
Cost	7628

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 10
Accuracy	73.4%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+222}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\frac{-0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{1}{x \cdot n}\right)\\ \end{array} \]

Alternative 11
Accuracy	73.1%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+219}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\frac{-0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{1}{x \cdot n}\right)\\ \end{array} \]

Alternative 12
Accuracy	44.7%
Cost	840

\[\begin{array}{l} \mathbf{if}\;n \leq -7200:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-283}:\\ \;\;\;\;-1 + \left(1 + \frac{1}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 13
Accuracy	36.6%
Cost	320

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

Alternative 14
Accuracy	37.2%
Cost	320

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

Alternative 15
Accuracy	4.6%
Cost	192

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

Reproduce

herbie shell --seed 2023137 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))