2nthrt (problem 3.4.6)

Average Accuracy: 49.2% → 98.1%

Time: 29.0s

Precision: binary64

Cost: 20488

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(0.3333333333333333 \cdot \frac{\mathsf{log1p}\left(x\right)}{n}\right) \cdot 3} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-10)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 5e-14)
     (/ (log1p (/ 1.0 x)) n)
     (-
      (exp (* (* 0.3333333333333333 (/ (log1p x) n)) 3.0))
      (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));
}

↓

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-10) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-14) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = exp(((0.3333333333333333 * (log1p(x) / n)) * 3.0)) - pow(x, (1.0 / 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 ((1.0 / n) <= -5e-10) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-14) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = Math.exp(((0.3333333333333333 * (Math.log1p(x) / n)) * 3.0)) - Math.pow(x, (1.0 / 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 (1.0 / n) <= -5e-10:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 5e-14:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = math.exp(((0.3333333333333333 * (math.log1p(x) / n)) * 3.0)) - math.pow(x, (1.0 / 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 (Float64(1.0 / n) <= -5e-10)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-14)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(exp(Float64(Float64(0.3333333333333333 * Float64(log1p(x) / n)) * 3.0)) - (x ^ Float64(1.0 / 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[N[(1.0 / n), $MachinePrecision], -5e-10], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-14], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[(0.3333333333333333 * N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -5.00000000000000031e-10

   1. Initial program 96.9%

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

   2. Applied egg-rr 96.8%

      \[\leadsto \color{blue}{e^{\left(0.3333333333333333 \cdot \frac{\mathsf{log1p}\left(x\right)}{n}\right) \cdot 3}} - {x}^{\left(\frac{1}{n}\right)} \]
      Proof

      [Start] 96.9	\[ {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
      add-cube-cbrt [=>] 96.8	\[ \color{blue}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - {x}^{\left(\frac{1}{n}\right)} \]
      pow3 [=>] 96.8	\[ \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)} \]
      pow-to-exp [=>] 96.8	\[ \color{blue}{e^{\log \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot 3}} - {x}^{\left(\frac{1}{n}\right)} \]
      pow1/3 [=>] 96.8	\[ e^{\log \color{blue}{\left({\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{0.3333333333333333}\right)} \cdot 3} - {x}^{\left(\frac{1}{n}\right)} \]
      log-pow [=>] 96.8	\[ e^{\color{blue}{\left(0.3333333333333333 \cdot \log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot 3} - {x}^{\left(\frac{1}{n}\right)} \]
      pow-to-exp [=>] 96.8	\[ e^{\left(0.3333333333333333 \cdot \log \color{blue}{\left(e^{\log \left(x + 1\right) \cdot \frac{1}{n}}\right)}\right) \cdot 3} - {x}^{\left(\frac{1}{n}\right)} \]
      add-log-exp [<=] 96.8	\[ e^{\left(0.3333333333333333 \cdot \color{blue}{\left(\log \left(x + 1\right) \cdot \frac{1}{n}\right)}\right) \cdot 3} - {x}^{\left(\frac{1}{n}\right)} \]
      un-div-inv [=>] 96.8	\[ e^{\left(0.3333333333333333 \cdot \color{blue}{\frac{\log \left(x + 1\right)}{n}}\right) \cdot 3} - {x}^{\left(\frac{1}{n}\right)} \]
      +-commutative [=>] 96.8	\[ e^{\left(0.3333333333333333 \cdot \frac{\log \color{blue}{\left(1 + x\right)}}{n}\right) \cdot 3} - {x}^{\left(\frac{1}{n}\right)} \]
      log1p-udef [<=] 96.8	\[ e^{\left(0.3333333333333333 \cdot \frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}\right) \cdot 3} - {x}^{\left(\frac{1}{n}\right)} \]

   3. Taylor expanded in x around inf 97.5%

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

   4. Simplified 97.5%

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

      [Start] 97.5	\[ \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x} \]
      log-rec [=>] 97.5	\[ \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]
      mul-1-neg [<=] 97.5	\[ \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]
      associate-*r/ [=>] 97.5	\[ \frac{e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}}}{n \cdot x} \]
      neg-mul-1 [<=] 97.5	\[ \frac{e^{\frac{\color{blue}{--1 \cdot \log x}}{n}}}{n \cdot x} \]
      mul-1-neg [=>] 97.5	\[ \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]
      remove-double-neg [=>] 97.5	\[ \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]
      *-commutative [=>] 97.5	\[ \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   if -5.00000000000000031e-10 < (/.f64 1 n) < 5.0000000000000002e-14

   1. Initial program 30.1%

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

   2. Taylor expanded in n around inf 77.5%

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

   3. Simplified 77.5%

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

      [Start] 77.5	\[ \frac{\log \left(1 + x\right) - \log x}{n} \]
      log1p-def [=>] 77.5	\[ \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Applied egg-rr 77.6%

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

      [Start] 77.5	\[ \frac{\mathsf{log1p}\left(x\right) - \log x}{n} \]
      log1p-udef [=>] 77.5	\[ \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]
      diff-log [=>] 77.6	\[ \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]
      +-commutative [=>] 77.6	\[ \frac{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}{n} \]

   5. Taylor expanded in n around 0 77.6%

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

   6. Simplified 99.3%

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

      [Start] 77.6	\[ \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]
      log-div [=>] 77.5	\[ \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]
      log1p-def [=>] 77.5	\[ \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]
      log1p-def [<=] 77.5	\[ \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]
      log-div [<=] 77.6	\[ \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]
      +-commutative [<=] 77.6	\[ \frac{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}{n} \]
      *-lft-identity [<=] 77.6	\[ \frac{\log \left(\frac{\color{blue}{1 \cdot \left(x + 1\right)}}{x}\right)}{n} \]
      associate-*l/ [<=] 74.0	\[ \frac{\log \color{blue}{\left(\frac{1}{x} \cdot \left(x + 1\right)\right)}}{n} \]
      distribute-rgt-in [=>] 74.0	\[ \frac{\log \color{blue}{\left(x \cdot \frac{1}{x} + 1 \cdot \frac{1}{x}\right)}}{n} \]
      *-lft-identity [=>] 74.0	\[ \frac{\log \left(x \cdot \frac{1}{x} + \color{blue}{\frac{1}{x}}\right)}{n} \]
      rgt-mult-inverse [=>] 77.6	\[ \frac{\log \left(\color{blue}{1} + \frac{1}{x}\right)}{n} \]
      log1p-def [=>] 99.3	\[ \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x}\right)}}{n} \]

   if 5.0000000000000002e-14 < (/.f64 1 n)

   1. Initial program 86.3%

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

   2. Applied egg-rr 91.3%

      \[\leadsto \color{blue}{e^{\left(0.3333333333333333 \cdot \frac{\mathsf{log1p}\left(x\right)}{n}\right) \cdot 3}} - {x}^{\left(\frac{1}{n}\right)} \]
      Proof

      [Start] 86.3	\[ {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
      add-cube-cbrt [=>] 86.1	\[ \color{blue}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - {x}^{\left(\frac{1}{n}\right)} \]
      pow3 [=>] 86.1	\[ \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)} \]
      pow-to-exp [=>] 86.1	\[ \color{blue}{e^{\log \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot 3}} - {x}^{\left(\frac{1}{n}\right)} \]
      pow1/3 [=>] 86.1	\[ e^{\log \color{blue}{\left({\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{0.3333333333333333}\right)} \cdot 3} - {x}^{\left(\frac{1}{n}\right)} \]
      log-pow [=>] 86.2	\[ e^{\color{blue}{\left(0.3333333333333333 \cdot \log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot 3} - {x}^{\left(\frac{1}{n}\right)} \]
      pow-to-exp [=>] 86.2	\[ e^{\left(0.3333333333333333 \cdot \log \color{blue}{\left(e^{\log \left(x + 1\right) \cdot \frac{1}{n}}\right)}\right) \cdot 3} - {x}^{\left(\frac{1}{n}\right)} \]
      add-log-exp [<=] 86.2	\[ e^{\left(0.3333333333333333 \cdot \color{blue}{\left(\log \left(x + 1\right) \cdot \frac{1}{n}\right)}\right) \cdot 3} - {x}^{\left(\frac{1}{n}\right)} \]
      un-div-inv [=>] 86.2	\[ e^{\left(0.3333333333333333 \cdot \color{blue}{\frac{\log \left(x + 1\right)}{n}}\right) \cdot 3} - {x}^{\left(\frac{1}{n}\right)} \]
      +-commutative [=>] 86.2	\[ e^{\left(0.3333333333333333 \cdot \frac{\log \color{blue}{\left(1 + x\right)}}{n}\right) \cdot 3} - {x}^{\left(\frac{1}{n}\right)} \]
      log1p-udef [<=] 91.3	\[ e^{\left(0.3333333333333333 \cdot \frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}\right) \cdot 3} - {x}^{\left(\frac{1}{n}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(0.3333333333333333 \cdot \frac{\mathsf{log1p}\left(x\right)}{n}\right) \cdot 3} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.1%
Cost	20232

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2
Accuracy	97.5%
Cost	14024

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 3
Accuracy	97.4%
Cost	13576

\[\begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{e^{t_0}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{expm1}\left(t_0\right)\\ \end{array} \]

Alternative 4
Accuracy	97.5%
Cost	13452

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{if}\;n \leq -7000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{+14}:\\ \;\;\;\;-\mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	97.8%
Cost	8076

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -12500000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\frac{t_1}{n \cdot x}\\ \mathbf{elif}\;n \leq 72000000:\\ \;\;\;\;\left(\left(1 + \frac{x}{n}\right) + 0.5 \cdot \frac{\frac{x \cdot x}{n}}{n}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	97.8%
Cost	7436

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -11600000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\frac{t_1}{n \cdot x}\\ \mathbf{elif}\;n \leq 28500000:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Accuracy	97.4%
Cost	7180

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{if}\;n \leq -7.9:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-308}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 2300000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Accuracy	97.6%
Cost	7180

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -5500000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\frac{t_1}{n \cdot x}\\ \mathbf{elif}\;n \leq 32000000:\\ \;\;\;\;1 - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Accuracy	89.0%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;n \leq -7.8 \lor \neg \left(n \leq -5 \cdot 10^{-308}\right):\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10
Accuracy	74.9%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 4.55 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+187}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+200}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Accuracy	54.5%
Cost	585

\[\begin{array}{l} \mathbf{if}\;n \leq -4.2 \lor \neg \left(n \leq 1.3 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12
Accuracy	55.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;n \leq -5.8 \lor \neg \left(n \leq 1.15 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13
Accuracy	55.2%
Cost	584

\[\begin{array}{l} \mathbf{if}\;n \leq -8:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-39}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 14
Accuracy	39.3%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))