2nthrt (problem 3.4.6)

Average Error: 32.8 → 1.3

Time: 29.2s

Precision: binary64

Cost: 13188

Math

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

↓

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

FPCore

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

↓

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (- (expm1 (/ (log x) n)))
   (* (/ 1.0 n) (/ (pow x (/ 1.0 n)) 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 tmp;
	if (x <= 1.0) {
		tmp = -expm1((log(x) / n));
	} else {
		tmp = (1.0 / n) * (pow(x, (1.0 / n)) / x);
	}
	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 <= 1.0) {
		tmp = -Math.expm1((Math.log(x) / n));
	} else {
		tmp = (1.0 / n) * (Math.pow(x, (1.0 / n)) / x);
	}
	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 <= 1.0:
		tmp = -math.expm1((math.log(x) / n))
	else:
		tmp = (1.0 / n) * (math.pow(x, (1.0 / n)) / 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)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-expm1(Float64(log(x) / n)));
	else
		tmp = Float64(Float64(1.0 / n) * Float64((x ^ Float64(1.0 / n)) / 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_] := If[LessEqual[x, 1.0], (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision]), N[(N[(1.0 / n), $MachinePrecision] * N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 47.3

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

   2. Taylor expanded in x around 0 47.4

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

   3. Taylor expanded in x around 0 47.4

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

   4. Simplified 1.7

      \[\leadsto \color{blue}{-\mathsf{expm1}\left(\frac{\log x}{n}\right)} \]
      Proof
      (neg.f64 (expm1.f64 (/.f64 (log.f64 x) n))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (/.f64 (log.f64 x) n)) 1))): 102 points increase in error, 75 points decrease in error
      (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (exp.f64 (/.f64 (log.f64 x) n)) 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= metadata-eval (log.f64 1)) (-.f64 (exp.f64 (/.f64 (log.f64 x) n)) 1)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 (log.f64 1) (exp.f64 (/.f64 (log.f64 x) n))) 1)): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (Rewrite=> metadata-eval 0) (exp.f64 (/.f64 (log.f64 x) n))) 1): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (exp.f64 (/.f64 (log.f64 x) n)))) 1): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 1 (neg.f64 (exp.f64 (/.f64 (log.f64 x) n))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 1 (exp.f64 (/.f64 (log.f64 x) n)))): 0 points increase in error, 0 points decrease in error

   if 1 < x

   1. Initial program 20.7

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

   2. Taylor expanded in x around inf 12.2

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

   3. Simplified 12.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{\log x}{n}}}{x \cdot x}, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{e^{\frac{\log x}{n}}}{x \cdot n}\right)} \]
      Proof
      (fma.f64 (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x x)) (+.f64 (/.f64 1/2 (*.f64 n n)) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (log.f64 x)))) n)) (*.f64 x x)) (+.f64 (/.f64 1/2 (*.f64 n n)) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (/.f64 (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 x)))) n)) (*.f64 x x)) (+.f64 (/.f64 1/2 (*.f64 n n)) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (*.f64 -1 (log.f64 x)) n)))) (*.f64 x x)) (+.f64 (/.f64 1/2 (*.f64 n n)) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (neg.f64 (/.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 (log.f64 x))) n))) (*.f64 x x)) (+.f64 (/.f64 1/2 (*.f64 n n)) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (neg.f64 (/.f64 (Rewrite<= log-rec_binary64 (log.f64 (/.f64 1 x))) n))) (*.f64 x x)) (+.f64 (/.f64 1/2 (*.f64 n n)) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n)))) (*.f64 x x)) (+.f64 (/.f64 1/2 (*.f64 n n)) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (Rewrite<= unpow2_binary64 (pow.f64 x 2))) (+.f64 (/.f64 1/2 (*.f64 n n)) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) (*.f64 n n)) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (+.f64 (/.f64 (*.f64 1/2 1) (Rewrite<= unpow2_binary64 (pow.f64 n 2))) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2)))) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (/.f64 (Rewrite<= metadata-eval (neg.f64 1/2)) n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1/2 n)))) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (neg.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) n))) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (neg.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 n))))) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n)))) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n))) (/.f64 (exp.f64 (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (log.f64 x)))) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n))) (/.f64 (exp.f64 (/.f64 (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 x)))) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n))) (/.f64 (exp.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (*.f64 -1 (log.f64 x)) n)))) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n))) (/.f64 (exp.f64 (neg.f64 (/.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 (log.f64 x))) n))) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n))) (/.f64 (exp.f64 (neg.f64 (/.f64 (Rewrite<= log-rec_binary64 (log.f64 (/.f64 1 x))) n))) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n))) (/.f64 (exp.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n)))) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n))) (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (Rewrite<= *-commutative_binary64 (*.f64 n x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n)))) (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (*.f64 n x)))): 2 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n)))) (pow.f64 x 2))) (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (*.f64 n x))): 5 points increase in error, 10 points decrease in error

   4. Taylor expanded in x around inf 1.8

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

   5. Simplified 1.8

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
      Proof
      (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n)): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 (log.f64 x) n))))) (*.f64 x n)): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (neg.f64 (Rewrite<= distribute-frac-neg_binary64 (/.f64 (neg.f64 (log.f64 x)) n)))) (*.f64 x n)): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (neg.f64 (/.f64 (Rewrite<= log-rec_binary64 (log.f64 (/.f64 1 x))) n))) (*.f64 x n)): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n)))) (*.f64 x n)): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (Rewrite<= *-commutative_binary64 (*.f64 n x))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 1.0

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x} \cdot \frac{1}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.3

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

Alternatives

Alternative 1
Error	6.9
Cost	7436

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.12 \cdot 10^{-254}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-245}:\\ \;\;\;\;1 - t_0\\ \mathbf{elif}\;x \leq 175:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{t_0}{x}\\ \end{array} \]

Alternative 2
Error	7.4
Cost	7308

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.12 \cdot 10^{-254}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-245}:\\ \;\;\;\;1 - t_0\\ \mathbf{elif}\;x \leq 175:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot n}\\ \end{array} \]

Alternative 3
Error	16.0
Cost	7116

\[\begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{-254}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-245}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} \cdot \left(\left(1 + \frac{0.3333333333333333}{x \cdot x}\right) + \frac{-0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 4
Error	12.2
Cost	7108

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

Alternative 5
Error	15.6
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} \cdot \left(\left(1 + \frac{0.3333333333333333}{x \cdot x}\right) + \frac{-0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 6
Error	15.8
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} \cdot \left(\left(1 + \frac{0.3333333333333333}{x \cdot x}\right) + \frac{-0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 7
Error	35.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -20:\\ \;\;\;\;\left(1 + \frac{1}{x \cdot n}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 8
Error	28.7
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{1}{n}}{x}\\ \mathbf{if}\;n \leq -0.6642624376975145:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-102}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	40.0
Cost	320

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

Alternative 10
Error	40.0
Cost	320

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

Alternative 11
Error	61.1
Cost	192

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

Reproduce

herbie shell --seed 2022316 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))