2nthrt (problem 3.4.6)

Percentage Accurate: 54.2% → 79.1%

Time: 22.3s

Precision: binary64

Cost: 39756

• could not determine a ground truth

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\log x}{n}\\ t_1 := \mathsf{fma}\left(-0.16666666666666666, {t_0}^{3}, -0.5 \cdot {t_0}^{2}\right) - t_0\\ \mathbf{if}\;x \leq 9.6 \cdot 10^{-147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \log \left(\sqrt{e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;x \leq 0.32:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;{x}^{\left({n}^{-1}\right)} \cdot \frac{{n}^{-1}}{x}\\ \end{array} \]

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) n))
        (t_1
         (-
          (fma -0.16666666666666666 (pow t_0 3.0) (* -0.5 (pow t_0 2.0)))
          t_0)))
   (if (<= x 9.6e-147)
     t_1
     (if (<= x 6e-93)
       (* 2.0 (log (sqrt (exp (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))))))
       (if (<= x 0.32) t_1 (* (pow x (pow n -1.0)) (/ (pow n -1.0) x)))))))

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) / n;
	double t_1 = fma(-0.16666666666666666, pow(t_0, 3.0), (-0.5 * pow(t_0, 2.0))) - t_0;
	double tmp;
	if (x <= 9.6e-147) {
		tmp = t_1;
	} else if (x <= 6e-93) {
		tmp = 2.0 * log(sqrt(exp((exp((log1p(x) / n)) - pow(x, (1.0 / n))))));
	} else if (x <= 0.32) {
		tmp = t_1;
	} else {
		tmp = pow(x, pow(n, -1.0)) * (pow(n, -1.0) / x);
	}
	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 = Float64(log(x) / n)
	t_1 = Float64(fma(-0.16666666666666666, (t_0 ^ 3.0), Float64(-0.5 * (t_0 ^ 2.0))) - t_0)
	tmp = 0.0
	if (x <= 9.6e-147)
		tmp = t_1;
	elseif (x <= 6e-93)
		tmp = Float64(2.0 * log(sqrt(exp(Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)))))));
	elseif (x <= 0.32)
		tmp = t_1;
	else
		tmp = Float64((x ^ (n ^ -1.0)) * Float64((n ^ -1.0) / x));
	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[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.16666666666666666 * N[Power[t$95$0, 3.0], $MachinePrecision] + N[(-0.5 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[x, 9.6e-147], t$95$1, If[LessEqual[x, 6e-93], N[(2.0 * N[Log[N[Sqrt[N[Exp[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.32], t$95$1, N[(N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Power[n, -1.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.59999999999999994e-147 or 6.0000000000000003e-93 < x < 0.320000000000000007

   1. Initial program 42.6%

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

   2. Taylor expanded in x around 0 41.8%

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

   3. Simplified 41.8%

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

      [Start] 41.8%	\[ 1 - e^{\frac{\log x}{n}} \]
      *-rgt-identity [<=] 41.8%	\[ 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]
      associate-*r/ [<=] 41.8%	\[ 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]
      unpow-1 [<=] 41.8%	\[ 1 - e^{\log x \cdot \color{blue}{{n}^{-1}}} \]
      exp-to-pow [=>] 41.8%	\[ 1 - \color{blue}{{x}^{\left({n}^{-1}\right)}} \]
      unpow-1 [=>] 41.8%	\[ 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   4. Taylor expanded in n around inf 76.6%

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

   5. Simplified 76.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {\left(\frac{\log x}{n}\right)}^{3}, -0.5 \cdot {\left(\frac{\log x}{n}\right)}^{2}\right) - \frac{\log x}{n}} \]
      Step-by-step derivation

      [Start] 76.6%	\[ -1 \cdot \frac{\log x}{n} + \left(-0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + -0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right) \]
      neg-mul-1 [<=] 76.6%	\[ \color{blue}{\left(-\frac{\log x}{n}\right)} + \left(-0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + -0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right) \]
      +-commutative [=>] 76.6%	\[ \color{blue}{\left(-0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + -0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right) + \left(-\frac{\log x}{n}\right)} \]
      unsub-neg [=>] 76.6%	\[ \color{blue}{\left(-0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + -0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right) - \frac{\log x}{n}} \]

   if 9.59999999999999994e-147 < x < 6.0000000000000003e-93

   1. Initial program 53.6%

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

   2. Applied egg-rr 79.4%

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

      [Start] 53.6%	\[ {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
      add-log-exp [=>] 53.9%	\[ \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]
      add-sqr-sqrt [=>] 53.9%	\[ \log \color{blue}{\left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)} \]
      log-prod [=>] 53.9%	\[ \color{blue}{\log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)} \]
      pow-to-exp [=>] 53.9%	\[ \log \left(\sqrt{e^{\color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) \]
      un-div-inv [=>] 53.9%	\[ \log \left(\sqrt{e^{e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) \]
      +-commutative [=>] 53.9%	\[ \log \left(\sqrt{e^{e^{\frac{\log \color{blue}{\left(1 + x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) \]
      log1p-udef [<=] 79.4%	\[ \log \left(\sqrt{e^{e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) \]
      inv-pow [=>] 79.4%	\[ \log \left(\sqrt{e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left({n}^{-1}\right)}}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) \]

   3. Simplified 79.4%

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

      [Start] 79.4%	\[ \log \left(\sqrt{e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left({n}^{-1}\right)}}}\right) + \log \left(\sqrt{e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left({n}^{-1}\right)}}}\right) \]
      count-2 [=>] 79.4%	\[ \color{blue}{2 \cdot \log \left(\sqrt{e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left({n}^{-1}\right)}}}\right)} \]
      rem-square-sqrt [<=] 79.4%	\[ 2 \cdot \log \left(\sqrt{e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - \color{blue}{\sqrt{{x}^{\left({n}^{-1}\right)}} \cdot \sqrt{{x}^{\left({n}^{-1}\right)}}}}}\right) \]
      rem-square-sqrt [<=] 79.4%	\[ 2 \cdot \log \left(\sqrt{e^{\color{blue}{\sqrt{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} \cdot \sqrt{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}}} - \sqrt{{x}^{\left({n}^{-1}\right)}} \cdot \sqrt{{x}^{\left({n}^{-1}\right)}}}}\right) \]
      difference-of-squares [=>] 79.4%	\[ 2 \cdot \log \left(\sqrt{e^{\color{blue}{\left(\sqrt{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} + \sqrt{{x}^{\left({n}^{-1}\right)}}\right) \cdot \left(\sqrt{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - \sqrt{{x}^{\left({n}^{-1}\right)}}\right)}}}\right) \]

   if 0.320000000000000007 < x

   1. Initial program 65.5%

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

   2. Taylor expanded in x around inf 96.7%

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

   3. Simplified 96.7%

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

      [Start] 96.7%	\[ \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x} \]
      log-rec [=>] 96.7%	\[ \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]
      mul-1-neg [<=] 96.7%	\[ \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]
      mul-1-neg [=>] 96.7%	\[ \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]
      distribute-frac-neg [=>] 96.7%	\[ \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]
      neg-mul-1 [<=] 96.7%	\[ \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]
      remove-double-neg [=>] 96.7%	\[ \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]
      *-rgt-identity [<=] 96.7%	\[ \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]
      associate-*r/ [<=] 96.7%	\[ \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]
      unpow-1 [<=] 96.7%	\[ \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x} \]
      exp-to-pow [=>] 96.7%	\[ \frac{\color{blue}{{x}^{\left({n}^{-1}\right)}}}{n \cdot x} \]
      unpow-1 [=>] 96.7%	\[ \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]
      *-commutative [=>] 96.7%	\[ \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   4. Applied egg-rr 98.1%

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

      [Start] 96.7%	\[ \frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} \]
      div-inv [=>] 96.7%	\[ \color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{x \cdot n}} \]
      inv-pow [=>] 96.7%	\[ {x}^{\color{blue}{\left({n}^{-1}\right)}} \cdot \frac{1}{x \cdot n} \]
      *-commutative [=>] 96.7%	\[ {x}^{\left({n}^{-1}\right)} \cdot \frac{1}{\color{blue}{n \cdot x}} \]
      associate-/r* [=>] 98.1%	\[ {x}^{\left({n}^{-1}\right)} \cdot \color{blue}{\frac{\frac{1}{n}}{x}} \]
      inv-pow [=>] 98.1%	\[ {x}^{\left({n}^{-1}\right)} \cdot \frac{\color{blue}{{n}^{-1}}}{x} \]

3. Recombined 3 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.6 \cdot 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, {\left(\frac{\log x}{n}\right)}^{3}, -0.5 \cdot {\left(\frac{\log x}{n}\right)}^{2}\right) - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \log \left(\sqrt{e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;x \leq 0.32:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, {\left(\frac{\log x}{n}\right)}^{3}, -0.5 \cdot {\left(\frac{\log x}{n}\right)}^{2}\right) - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;{x}^{\left({n}^{-1}\right)} \cdot \frac{{n}^{-1}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	79.1%
Cost	39756

\[\begin{array}{l} t_0 := \frac{\log x}{n}\\ t_1 := \mathsf{fma}\left(-0.16666666666666666, {t_0}^{3}, -0.5 \cdot {t_0}^{2}\right) - t_0\\ \mathbf{if}\;x \leq 9.6 \cdot 10^{-147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \log \left(\sqrt{e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;x \leq 0.32:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;{x}^{\left({n}^{-1}\right)} \cdot \frac{{n}^{-1}}{x}\\ \end{array} \]

Alternative 2
Accuracy	87.5%
Cost	125444

\[\begin{array}{l} t_0 := {\log x}^{2}\\ t_1 := \log \left(1 + x\right)\\ \mathbf{if}\;n \leq -1.36:\\ \;\;\;\;\left(0.5 \cdot \frac{{t_1}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{t_1}^{4}}{{n}^{4}} + \left(\frac{t_1 - \log x}{n} - \frac{-0.16666666666666666 \cdot {t_1}^{3} - -0.16666666666666666 \cdot {\log x}^{3}}{{n}^{3}}\right)\right)\right) - \left(0.5 \cdot \frac{t_0}{{n}^{2}} + 0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}}\right)\\ \mathbf{elif}\;n \leq 57000000:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\right) + \frac{t_0}{n \cdot n} \cdot -0.5\\ \end{array} \]

Alternative 3
Accuracy	87.5%
Cost	78980

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\right)\\ t_1 := \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\\ \mathbf{if}\;n \leq -1.36:\\ \;\;\;\;t_1 + \left(t_0 + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{{n}^{3}}\right)\\ \mathbf{elif}\;n \leq 32000000:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 + t_1\\ \end{array} \]

Alternative 4
Accuracy	87.5%
Cost	46409

\[\begin{array}{l} \mathbf{if}\;n \leq -4000000 \lor \neg \left(n \leq 36000000\right):\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 5
Accuracy	87.3%
Cost	19977

\[\begin{array}{l} \mathbf{if}\;n \leq -72000000000000 \lor \neg \left(n \leq 37000000000\right):\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 6
Accuracy	80.8%
Cost	8976

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{n \cdot \left(0.5 + x\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{1}{-n} \cdot \log \left(\frac{x}{1 + x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right) + \left(1 + \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 7
Accuracy	80.0%
Cost	8724

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{n \cdot \left(0.5 + x\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{1}{-n} \cdot \log \left(\frac{x}{1 + x}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+152}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}{n}\\ \end{array} \]

Alternative 8
Accuracy	78.3%
Cost	8080

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{n \cdot \left(0.5 + x\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{1}{-n} \cdot \log \left(\frac{x}{1 + x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - t_0\\ \end{array} \]

Alternative 9
Accuracy	78.3%
Cost	8080

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;t_0 \cdot \frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{n \cdot \left(0.5 + x\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{1}{-n} \cdot \log \left(\frac{x}{1 + x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - t_0\\ \end{array} \]

Alternative 10
Accuracy	78.5%
Cost	8080

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;t_0 \cdot \frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{n \cdot \left(0.5 + x\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{1}{-n} \cdot \log \left(\frac{x}{1 + x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \end{array} \]

Alternative 11
Accuracy	78.5%
Cost	8080

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{n \cdot \left(0.5 + x\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{1}{-n} \cdot \log \left(\frac{x}{1 + x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 12
Accuracy	78.3%
Cost	7888

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{n \cdot \left(0.5 + x\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 - t_0\\ \end{array} \]

Alternative 13
Accuracy	58.6%
Cost	7644

\[\begin{array}{l} t_0 := \frac{-\log x}{n}\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 7 \cdot 10^{-259}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{-181}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-147}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 0.84:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{1}{n}}{\left(x + \left(\frac{0.041666666666666664}{x \cdot x} + \frac{0.25}{x}\right)\right) + \left(0.5 + \frac{-0.3333333333333333}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 14
Accuracy	64.8%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{-134} \lor \neg \left(n \leq 8800000000\right):\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 15
Accuracy	60.2%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 0.84:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{1}{n}}{\left(x + \left(\frac{0.041666666666666664}{x \cdot x} + \frac{0.25}{x}\right)\right) + \left(0.5 + \frac{-0.3333333333333333}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 16
Accuracy	59.9%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.35:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{1}{n}}{\left(x + \left(\frac{0.041666666666666664}{x \cdot x} + \frac{0.25}{x}\right)\right) + \left(0.5 + \frac{-0.3333333333333333}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 17
Accuracy	44.7%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 18
Accuracy	39.9%
Cost	320

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

Alternative 19
Accuracy	40.5%
Cost	320

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

Alternative 20
Accuracy	40.5%
Cost	320

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

Reproduce

herbie shell --seed 2023171 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))