2nthrt (problem 3.4.6)

Average Error: 33.2 → 1.9

Time: 25.6s

Precision: binary64

Cost: 47108

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 0.56:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(\frac{\log x}{\frac{n}{6}}\right)}{-1 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}}{{\left(x \cdot \left(x + 1\right)\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{2}{n}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]

FPCore

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

↓

(FPCore (x n)
 :precision binary64
 (if (<= x 0.56)
   (/
    (/ (expm1 (/ (log x) (/ n 6.0))) (- -1.0 (pow (pow x (/ 1.0 n)) 3.0)))
    (+
     (pow (* x (+ x 1.0)) (/ 1.0 n))
     (+ (pow x (/ 2.0 n)) (pow (+ x 1.0) (/ 2.0 n)))))
   (/ (exp (/ (log x) n)) (* 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 tmp;
	if (x <= 0.56) {
		tmp = (expm1((log(x) / (n / 6.0))) / (-1.0 - pow(pow(x, (1.0 / n)), 3.0))) / (pow((x * (x + 1.0)), (1.0 / n)) + (pow(x, (2.0 / n)) + pow((x + 1.0), (2.0 / n))));
	} else {
		tmp = exp((log(x) / n)) / (x * n);
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.56) {
		tmp = (Math.expm1((Math.log(x) / (n / 6.0))) / (-1.0 - Math.pow(Math.pow(x, (1.0 / n)), 3.0))) / (Math.pow((x * (x + 1.0)), (1.0 / n)) + (Math.pow(x, (2.0 / n)) + Math.pow((x + 1.0), (2.0 / n))));
	} else {
		tmp = Math.exp((Math.log(x) / n)) / (x * n);
	}
	return tmp;
}

Python

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

↓

def code(x, n):
	tmp = 0
	if x <= 0.56:
		tmp = (math.expm1((math.log(x) / (n / 6.0))) / (-1.0 - math.pow(math.pow(x, (1.0 / n)), 3.0))) / (math.pow((x * (x + 1.0)), (1.0 / n)) + (math.pow(x, (2.0 / n)) + math.pow((x + 1.0), (2.0 / n))))
	else:
		tmp = math.exp((math.log(x) / n)) / (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)
	tmp = 0.0
	if (x <= 0.56)
		tmp = Float64(Float64(expm1(Float64(log(x) / Float64(n / 6.0))) / Float64(-1.0 - ((x ^ Float64(1.0 / n)) ^ 3.0))) / Float64((Float64(x * Float64(x + 1.0)) ^ Float64(1.0 / n)) + Float64((x ^ Float64(2.0 / n)) + (Float64(x + 1.0) ^ Float64(2.0 / n)))));
	else
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(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_] := If[LessEqual[x, 0.56], N[(N[(N[(Exp[N[(N[Log[x], $MachinePrecision] / N[(n / 6.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(-1.0 - N[Power[N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] + N[(N[Power[x, N[(2.0 / n), $MachinePrecision]], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], N[(2.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.56000000000000005

   1. Initial program 47.4

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

   2. Applied egg-rr 47.4

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

   3. Simplified 47.4

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

      [Start] 47.4	\[ \left({\left(x + 1\right)}^{\left(\frac{3}{n}\right)} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(x + x \cdot x\right)}^{\left({n}^{-1}\right)} + \left({\left(x + 1\right)}^{\left(\frac{2}{n}\right)} + {x}^{\left(\frac{2}{n}\right)}\right)} \]
      associate-*r/ [=>] 47.4	\[ \color{blue}{\frac{\left({\left(x + 1\right)}^{\left(\frac{3}{n}\right)} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot 1}{{\left(x + x \cdot x\right)}^{\left({n}^{-1}\right)} + \left({\left(x + 1\right)}^{\left(\frac{2}{n}\right)} + {x}^{\left(\frac{2}{n}\right)}\right)}} \]
      +-commutative [=>] 47.4	\[ \frac{\left({\color{blue}{\left(1 + x\right)}}^{\left(\frac{3}{n}\right)} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot 1}{{\left(x + x \cdot x\right)}^{\left({n}^{-1}\right)} + \left({\left(x + 1\right)}^{\left(\frac{2}{n}\right)} + {x}^{\left(\frac{2}{n}\right)}\right)} \]
      distribute-rgt1-in [=>] 47.4	\[ \frac{\left({\left(1 + x\right)}^{\left(\frac{3}{n}\right)} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot 1}{{\color{blue}{\left(\left(x + 1\right) \cdot x\right)}}^{\left({n}^{-1}\right)} + \left({\left(x + 1\right)}^{\left(\frac{2}{n}\right)} + {x}^{\left(\frac{2}{n}\right)}\right)} \]
      +-commutative [=>] 47.4	\[ \frac{\left({\left(1 + x\right)}^{\left(\frac{3}{n}\right)} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot 1}{{\left(\color{blue}{\left(1 + x\right)} \cdot x\right)}^{\left({n}^{-1}\right)} + \left({\left(x + 1\right)}^{\left(\frac{2}{n}\right)} + {x}^{\left(\frac{2}{n}\right)}\right)} \]
      unpow-1 [=>] 47.4	\[ \frac{\left({\left(1 + x\right)}^{\left(\frac{3}{n}\right)} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\color{blue}{\left(\frac{1}{n}\right)}} + \left({\left(x + 1\right)}^{\left(\frac{2}{n}\right)} + {x}^{\left(\frac{2}{n}\right)}\right)} \]
      +-commutative [=>] 47.4	\[ \frac{\left({\left(1 + x\right)}^{\left(\frac{3}{n}\right)} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \color{blue}{\left({x}^{\left(\frac{2}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{2}{n}\right)}\right)}} \]
      +-commutative [=>] 47.4	\[ \frac{\left({\left(1 + x\right)}^{\left(\frac{3}{n}\right)} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\color{blue}{\left(1 + x\right)}}^{\left(\frac{2}{n}\right)}\right)} \]

   4. Taylor expanded in x around 0 47.4

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

   5. Applied egg-rr 47.4

      \[\leadsto \frac{\color{blue}{\frac{-\left(1 - e^{\frac{\log x}{n} \cdot 6}\right)}{-\left(1 + {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}} \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{2}{n}\right)}\right)} \]

   6. Simplified 1.9

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(\frac{\log x}{\frac{n}{6}}\right)}{-1 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}} \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{2}{n}\right)}\right)} \]
      Proof

      [Start] 47.4	\[ \frac{\frac{-\left(1 - e^{\frac{\log x}{n} \cdot 6}\right)}{-\left(1 + {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)} \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{2}{n}\right)}\right)} \]
      neg-sub0 [=>] 47.4	\[ \frac{\frac{\color{blue}{0 - \left(1 - e^{\frac{\log x}{n} \cdot 6}\right)}}{-\left(1 + {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)} \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{2}{n}\right)}\right)} \]
      associate--r- [=>] 47.4	\[ \frac{\frac{\color{blue}{\left(0 - 1\right) + e^{\frac{\log x}{n} \cdot 6}}}{-\left(1 + {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)} \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{2}{n}\right)}\right)} \]
      metadata-eval [=>] 47.4	\[ \frac{\frac{\color{blue}{-1} + e^{\frac{\log x}{n} \cdot 6}}{-\left(1 + {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)} \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{2}{n}\right)}\right)} \]
      +-commutative [<=] 47.4	\[ \frac{\frac{\color{blue}{e^{\frac{\log x}{n} \cdot 6} + -1}}{-\left(1 + {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)} \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{2}{n}\right)}\right)} \]
      metadata-eval [<=] 47.4	\[ \frac{\frac{e^{\frac{\log x}{n} \cdot 6} + \color{blue}{\left(-1\right)}}{-\left(1 + {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)} \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{2}{n}\right)}\right)} \]
      sub-neg [<=] 47.4	\[ \frac{\frac{\color{blue}{e^{\frac{\log x}{n} \cdot 6} - 1}}{-\left(1 + {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)} \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{2}{n}\right)}\right)} \]
      expm1-def [=>] 1.9	\[ \frac{\frac{\color{blue}{\mathsf{expm1}\left(\frac{\log x}{n} \cdot 6\right)}}{-\left(1 + {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)} \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{2}{n}\right)}\right)} \]
      associate-*l/ [=>] 1.9	\[ \frac{\frac{\mathsf{expm1}\left(\color{blue}{\frac{\log x \cdot 6}{n}}\right)}{-\left(1 + {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)} \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{2}{n}\right)}\right)} \]
      associate-/l* [=>] 1.9	\[ \frac{\frac{\mathsf{expm1}\left(\color{blue}{\frac{\log x}{\frac{n}{6}}}\right)}{-\left(1 + {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)} \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{2}{n}\right)}\right)} \]
      neg-sub0 [=>] 1.9	\[ \frac{\frac{\mathsf{expm1}\left(\frac{\log x}{\frac{n}{6}}\right)}{\color{blue}{0 - \left(1 + {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}} \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{2}{n}\right)}\right)} \]
      associate--r+ [=>] 1.9	\[ \frac{\frac{\mathsf{expm1}\left(\frac{\log x}{\frac{n}{6}}\right)}{\color{blue}{\left(0 - 1\right) - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}} \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{2}{n}\right)}\right)} \]
      metadata-eval [=>] 1.9	\[ \frac{\frac{\mathsf{expm1}\left(\frac{\log x}{\frac{n}{6}}\right)}{\color{blue}{-1} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}} \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{2}{n}\right)}\right)} \]

   if 0.56000000000000005 < x

   1. Initial program 21.1

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

   2. Taylor expanded in x around inf 1.8

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

   3. Simplified 1.8

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

      [Start] 1.8	\[ \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x} \]
      mul-1-neg [=>] 1.8	\[ \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]
      log-rec [=>] 1.8	\[ \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]
      mul-1-neg [<=] 1.8	\[ \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]
      distribute-neg-frac [=>] 1.8	\[ \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]
      mul-1-neg [=>] 1.8	\[ \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]
      remove-double-neg [=>] 1.8	\[ \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]
      *-commutative [=>] 1.8	\[ \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.56:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(\frac{\log x}{\frac{n}{6}}\right)}{-1 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}}{{\left(x \cdot \left(x + 1\right)\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{2}{n}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.3
Cost	14420

\[\begin{array}{l} t_0 := \frac{\log x}{n}\\ t_1 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -20000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -1.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{1}{n} \cdot \left(1 + t_0\right)}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 - e^{t_0}\\ \end{array} \]

Alternative 2
Error	13.2
Cost	14420

\[\begin{array}{l} t_0 := \frac{\log x}{n}\\ t_1 := \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -20000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -1.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{1}{n} \cdot \left(1 + t_0\right)}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-90}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 - e^{t_0}\\ \end{array} \]

Alternative 3
Error	7.4
Cost	13644

\[\begin{array}{l} t_0 := \frac{\log x}{n}\\ t_1 := e^{t_0}\\ \mathbf{if}\;x \leq 8 \cdot 10^{-295}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-285}:\\ \;\;\;\;1 - t_1\\ \mathbf{elif}\;x \leq 8500:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{x \cdot n}\\ \end{array} \]

Alternative 4
Error	7.4
Cost	13644

\[\begin{array}{l} t_0 := \frac{\log x}{n}\\ t_1 := e^{t_0}\\ \mathbf{if}\;x \leq 8 \cdot 10^{-295}:\\ \;\;\;\;\frac{\frac{1}{n}}{\frac{1}{\mathsf{log1p}\left(x\right) - \log x}}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-285}:\\ \;\;\;\;1 - t_1\\ \mathbf{elif}\;x \leq 48000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{x \cdot n}\\ \end{array} \]

Alternative 5
Error	14.1
Cost	8336

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;n \leq -5.4 \cdot 10^{+73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -450:\\ \;\;\;\;\frac{\frac{1}{n} \cdot \left(1 + \frac{\log x}{n}\right)}{x}\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 7500:\\ \;\;\;\;1 + \left(\left(\frac{\frac{0.5}{n} + -0.5}{n} \cdot \left(x \cdot x\right) + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{+56} \lor \neg \left(n \leq 1.85 \cdot 10^{+87}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 6
Error	16.2
Cost	7369

\[\begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+28} \lor \neg \left(x \leq 2.8 \cdot 10^{+139}\right):\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1 + \frac{\log x}{n}}{x}\\ \end{array} \]

Alternative 7
Error	16.2
Cost	7241

\[\begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+28} \lor \neg \left(x \leq 2.4 \cdot 10^{+139}\right):\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\log x}{n}}{n}}{x}\\ \end{array} \]

Alternative 8
Error	16.2
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+28} \lor \neg \left(x \leq 3 \cdot 10^{+139}\right):\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 9
Error	16.9
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+179}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{-0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{1}{x \cdot n}\right)\\ \end{array} \]

Alternative 10
Error	17.1
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+179}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{-0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{1}{x \cdot n}\right)\\ \end{array} \]

Alternative 11
Error	35.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;n \leq -3.3 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-262}:\\ \;\;\;\;-1 + \left(1 + \frac{1}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 12
Error	40.8
Cost	320

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

Alternative 13
Error	40.4
Cost	320

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

Alternative 14
Error	61.1
Cost	192

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

Reproduce

herbie shell --seed 2023039 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))