2nthrt (problem 3.4.6)

Average Error: 33.2 → 1.9

Time: 25.2s

Precision: binary64

Cost: 47108

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(t_0 \cdot 6\right)}{-1 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}}{{\left(x \cdot \left(x + 1\right)\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{2}{n}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{t_0}}{x \cdot n}\\ \end{array} \]

FPCore

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

↓

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= x 0.55)
     (/
      (/ (expm1 (* t_0 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 t_0) (* 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) / n;
	double tmp;
	if (x <= 0.55) {
		tmp = (expm1((t_0 * 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(t_0) / (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 t_0 = Math.log(x) / n;
	double tmp;
	if (x <= 0.55) {
		tmp = (Math.expm1((t_0 * 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(t_0) / (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):
	t_0 = math.log(x) / n
	tmp = 0
	if x <= 0.55:
		tmp = (math.expm1((t_0 * 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(t_0) / (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 = Float64(log(x) / n)
	tmp = 0.0
	if (x <= 0.55)
		tmp = Float64(Float64(expm1(Float64(t_0 * 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(t_0) / 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_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[x, 0.55], N[(N[(N[(Exp[N[(t$95$0 * 6.0), $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[t$95$0], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.55000000000000004

   1. Initial program 47.6

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

   2. Applied egg-rr 47.6

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

   3. Simplified 47.6

      \[\leadsto \color{blue}{\frac{\left({\left(1 + x\right)}^{\left(\frac{3}{n}\right)} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{2}{n}\right)}\right)}} \]
      Proof
      (/.f64 (*.f64 (-.f64 (pow.f64 (+.f64 1 x) (/.f64 3 n)) (pow.f64 x (/.f64 3 n))) 1) (+.f64 (pow.f64 (*.f64 (+.f64 1 x) x) (/.f64 1 n)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (-.f64 (pow.f64 (Rewrite<= +-commutative_binary64 (+.f64 x 1)) (/.f64 3 n)) (pow.f64 x (/.f64 3 n))) 1) (+.f64 (pow.f64 (*.f64 (+.f64 1 x) x) (/.f64 1 n)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 8 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (-.f64 (pow.f64 (+.f64 x 1) (/.f64 3 n)) (pow.f64 x (/.f64 3 n))) 1) (+.f64 (pow.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 x 1)) x) (/.f64 1 n)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 0 points increase in error, 8 points decrease in error
      (/.f64 (*.f64 (-.f64 (pow.f64 (+.f64 x 1) (/.f64 3 n)) (pow.f64 x (/.f64 3 n))) 1) (+.f64 (pow.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 x (*.f64 x x))) (/.f64 1 n)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 8 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (-.f64 (pow.f64 (+.f64 x 1) (/.f64 3 n)) (pow.f64 x (/.f64 3 n))) 1) (+.f64 (pow.f64 (+.f64 x (*.f64 x x)) (Rewrite<= unpow-1_binary64 (pow.f64 n -1))) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 8 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (-.f64 (pow.f64 (+.f64 x 1) (/.f64 3 n)) (pow.f64 x (/.f64 3 n))) 1) (+.f64 (pow.f64 (+.f64 x (*.f64 x x)) (pow.f64 n -1)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (Rewrite<= +-commutative_binary64 (+.f64 x 1)) (/.f64 2 n))))): 0 points increase in error, 8 points decrease in error
      (/.f64 (*.f64 (-.f64 (pow.f64 (+.f64 x 1) (/.f64 3 n)) (pow.f64 x (/.f64 3 n))) 1) (+.f64 (pow.f64 (+.f64 x (*.f64 x x)) (pow.f64 n -1)) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (+.f64 x 1) (/.f64 2 n)) (pow.f64 x (/.f64 2 n)))))): 8 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (-.f64 (pow.f64 (+.f64 x 1) (/.f64 3 n)) (pow.f64 x (/.f64 3 n))) (/.f64 1 (+.f64 (pow.f64 (+.f64 x (*.f64 x x)) (pow.f64 n -1)) (+.f64 (pow.f64 (+.f64 x 1) (/.f64 2 n)) (pow.f64 x (/.f64 2 n))))))): 0 points increase in error, 8 points decrease in error

   4. Taylor expanded in x around 0 47.6

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

   5. Applied egg-rr 47.6

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

   6. Simplified 1.9

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(\frac{\log x}{n} \cdot 6\right)}{-1 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}} \cdot 1}{{\left(\left(1 + x\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{2}{n}\right)}\right)} \]
      Proof
      (/.f64 (*.f64 (/.f64 (expm1.f64 (*.f64 (/.f64 (log.f64 x) n) 6)) (-.f64 -1 (pow.f64 (pow.f64 x (/.f64 1 n)) 3))) 1) (+.f64 (pow.f64 (*.f64 (+.f64 1 x) x) (/.f64 1 n)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 (/.f64 (log.f64 x) n) 6)) 1)) (-.f64 -1 (pow.f64 (pow.f64 x (/.f64 1 n)) 3))) 1) (+.f64 (pow.f64 (*.f64 (+.f64 1 x) x) (/.f64 1 n)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 11 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (Rewrite=> sub-neg_binary64 (+.f64 (exp.f64 (*.f64 (/.f64 (log.f64 x) n) 6)) (neg.f64 1))) (-.f64 -1 (pow.f64 (pow.f64 x (/.f64 1 n)) 3))) 1) (+.f64 (pow.f64 (*.f64 (+.f64 1 x) x) (/.f64 1 n)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 0 points increase in error, 11 points decrease in error
      (/.f64 (*.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (log.f64 x) n) 6)) (Rewrite=> metadata-eval -1)) (-.f64 -1 (pow.f64 (pow.f64 x (/.f64 1 n)) 3))) 1) (+.f64 (pow.f64 (*.f64 (+.f64 1 x) x) (/.f64 1 n)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 11 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 -1 (exp.f64 (*.f64 (/.f64 (log.f64 x) n) 6)))) (-.f64 -1 (pow.f64 (pow.f64 x (/.f64 1 n)) 3))) 1) (+.f64 (pow.f64 (*.f64 (+.f64 1 x) x) (/.f64 1 n)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 10 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (+.f64 (Rewrite<= metadata-eval (-.f64 0 1)) (exp.f64 (*.f64 (/.f64 (log.f64 x) n) 6))) (-.f64 -1 (pow.f64 (pow.f64 x (/.f64 1 n)) 3))) 1) (+.f64 (pow.f64 (*.f64 (+.f64 1 x) x) (/.f64 1 n)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 1 points increase in error, 10 points decrease in error
      (/.f64 (*.f64 (/.f64 (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 1 (exp.f64 (*.f64 (/.f64 (log.f64 x) n) 6))))) (-.f64 -1 (pow.f64 (pow.f64 x (/.f64 1 n)) 3))) 1) (+.f64 (pow.f64 (*.f64 (+.f64 1 x) x) (/.f64 1 n)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 10 points increase in error, 1 points decrease in error
      (/.f64 (*.f64 (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 1 (exp.f64 (*.f64 (/.f64 (log.f64 x) n) 6))))) (-.f64 -1 (pow.f64 (pow.f64 x (/.f64 1 n)) 3))) 1) (+.f64 (pow.f64 (*.f64 (+.f64 1 x) x) (/.f64 1 n)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 1 points increase in error, 10 points decrease in error
      (/.f64 (*.f64 (/.f64 (neg.f64 (-.f64 1 (exp.f64 (*.f64 (/.f64 (log.f64 x) n) 6)))) (Rewrite<= unsub-neg_binary64 (+.f64 -1 (neg.f64 (pow.f64 (pow.f64 x (/.f64 1 n)) 3))))) 1) (+.f64 (pow.f64 (*.f64 (+.f64 1 x) x) (/.f64 1 n)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 0 points increase in error, 11 points decrease in error
      (/.f64 (*.f64 (/.f64 (neg.f64 (-.f64 1 (exp.f64 (*.f64 (/.f64 (log.f64 x) n) 6)))) (+.f64 (Rewrite<= metadata-eval (neg.f64 1)) (neg.f64 (pow.f64 (pow.f64 x (/.f64 1 n)) 3)))) 1) (+.f64 (pow.f64 (*.f64 (+.f64 1 x) x) (/.f64 1 n)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (neg.f64 (-.f64 1 (exp.f64 (*.f64 (/.f64 (log.f64 x) n) 6)))) (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 1 (pow.f64 (pow.f64 x (/.f64 1 n)) 3))))) 1) (+.f64 (pow.f64 (*.f64 (+.f64 1 x) x) (/.f64 1 n)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 11 points increase in error, 0 points decrease in error

   if 0.55000000000000004 < x

   1. Initial program 20.9

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

   2. Taylor expanded in x around inf 2.0

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

   3. Simplified 2.0

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
      Proof
      (/.f64 (*.f64 (-.f64 (pow.f64 (+.f64 1 x) (/.f64 3 n)) (pow.f64 x (/.f64 3 n))) 1) (+.f64 (pow.f64 (*.f64 (+.f64 1 x) x) (/.f64 1 n)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (-.f64 (pow.f64 (Rewrite<= +-commutative_binary64 (+.f64 x 1)) (/.f64 3 n)) (pow.f64 x (/.f64 3 n))) 1) (+.f64 (pow.f64 (*.f64 (+.f64 1 x) x) (/.f64 1 n)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 8 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (-.f64 (pow.f64 (+.f64 x 1) (/.f64 3 n)) (pow.f64 x (/.f64 3 n))) 1) (+.f64 (pow.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 x 1)) x) (/.f64 1 n)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 0 points increase in error, 8 points decrease in error
      (/.f64 (*.f64 (-.f64 (pow.f64 (+.f64 x 1) (/.f64 3 n)) (pow.f64 x (/.f64 3 n))) 1) (+.f64 (pow.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 x (*.f64 x x))) (/.f64 1 n)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 8 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (-.f64 (pow.f64 (+.f64 x 1) (/.f64 3 n)) (pow.f64 x (/.f64 3 n))) 1) (+.f64 (pow.f64 (+.f64 x (*.f64 x x)) (Rewrite<= unpow-1_binary64 (pow.f64 n -1))) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (+.f64 1 x) (/.f64 2 n))))): 8 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (-.f64 (pow.f64 (+.f64 x 1) (/.f64 3 n)) (pow.f64 x (/.f64 3 n))) 1) (+.f64 (pow.f64 (+.f64 x (*.f64 x x)) (pow.f64 n -1)) (+.f64 (pow.f64 x (/.f64 2 n)) (pow.f64 (Rewrite<= +-commutative_binary64 (+.f64 x 1)) (/.f64 2 n))))): 0 points increase in error, 8 points decrease in error
      (/.f64 (*.f64 (-.f64 (pow.f64 (+.f64 x 1) (/.f64 3 n)) (pow.f64 x (/.f64 3 n))) 1) (+.f64 (pow.f64 (+.f64 x (*.f64 x x)) (pow.f64 n -1)) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (+.f64 x 1) (/.f64 2 n)) (pow.f64 x (/.f64 2 n)))))): 8 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (-.f64 (pow.f64 (+.f64 x 1) (/.f64 3 n)) (pow.f64 x (/.f64 3 n))) (/.f64 1 (+.f64 (pow.f64 (+.f64 x (*.f64 x x)) (pow.f64 n -1)) (+.f64 (pow.f64 (+.f64 x 1) (/.f64 2 n)) (pow.f64 x (/.f64 2 n))))))): 0 points increase in error, 8 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(\frac{\log x}{n} \cdot 6\right)}{-1 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}}{{\left(x \cdot \left(x + 1\right)\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{2}{n}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.2
Cost	39244

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;n \leq -3.1 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -150000000:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;n \leq 1900000000:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left({n}^{-1}\right)}\right)\right)\\ \mathbf{elif}\;n \leq 2.2 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(n \cdot \frac{1}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}\\ \end{array} \]

Alternative 2
Error	12.1
Cost	20504

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;n \leq -5.4 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -2000000000:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;n \leq 64000000000:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 1.7 \cdot 10^{+40}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;n \leq 10^{+66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 4 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(n \cdot \frac{1}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}\\ \end{array} \]

Alternative 3
Error	12.2
Cost	20376

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;n \leq -2.6 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 4.350538613840888 \cdot 10^{-283}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;n \leq 1000000000:\\ \;\;\;\;\left(\frac{x}{n} + \left(-0.5 + \frac{0.5}{n}\right) \cdot \left(x \cdot \frac{x}{n}\right)\right) + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;n \leq 6.2 \cdot 10^{+40}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;n \leq 8.6 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}\\ \end{array} \]

Alternative 4
Error	12.1
Cost	20376

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;n \leq -6.2 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -2000000000:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;n \leq 4500000000000:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 4.7 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{+69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}\\ \end{array} \]

Alternative 5
Error	12.2
Cost	14168

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;n \leq -2.1 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 4.350538613840888 \cdot 10^{-283}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;n \leq 105000000000:\\ \;\;\;\;\left(\frac{x}{n} + \left(-0.5 + \frac{0.5}{n}\right) \cdot \left(x \cdot \frac{x}{n}\right)\right) + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{+66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{\mathsf{log1p}\left(x\right) - \log x}}}{n}\\ \end{array} \]

Alternative 6
Error	12.3
Cost	13508

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-52}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-21}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n} + \left(-0.5 + \frac{0.5}{n}\right) \cdot \left(x \cdot \frac{x}{n}\right)\right) + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \]

Alternative 7
Error	13.0
Cost	9368

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -50000000000000:\\ \;\;\;\;0\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-21}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n} + \left(-0.5 + \frac{0.5}{n}\right) \cdot \left(x \cdot \frac{x}{n}\right)\right) + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \]

Alternative 8
Error	13.1
Cost	8344

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -50000000000000:\\ \;\;\;\;0\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-21}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 9
Error	15.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 0.86:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+171}:\\ \;\;\;\;\frac{\left(\frac{1}{x} + \frac{1}{x} \cdot \frac{0.3333333333333333}{x \cdot x}\right) + \frac{-0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10
Error	15.9
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.62 \cdot 10^{+171}:\\ \;\;\;\;\frac{\left(\frac{1}{x} + \frac{1}{x} \cdot \frac{0.3333333333333333}{x \cdot x}\right) + \frac{-0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Error	29.7
Cost	585

\[\begin{array}{l} \mathbf{if}\;n \leq -24.5 \lor \neg \left(n \leq 1.15 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12
Error	29.3
Cost	585

\[\begin{array}{l} \mathbf{if}\;n \leq -5.4 \lor \neg \left(n \leq 2.5 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13
Error	29.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;n \leq -4.2:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{-67}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 14
Error	39.3
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022343 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))