2nthrt (problem 3.4.6)

Average Error: 32.8 → 7.4

Time: 37.0s

Precision: binary64

Cost: 67073

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 1.280852323654058 \cdot 10^{-211}:\\ \;\;\;\;-0.5 \cdot \frac{{\log x}^{2}}{n \cdot n} + \left(\left(\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3}\right) + \frac{\log \left(x + 1\right) - \log x}{n}\right) - 0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right)\\ \mathbf{elif}\;x \leq 2.2097631365031477 \cdot 10^{-202}:\\ \;\;\;\;\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\\ \mathbf{elif}\;x \leq 147972.98971215458:\\ \;\;\;\;-0.5 \cdot \frac{{\log x}^{2}}{n \cdot n} + \left(\left(\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3}\right) + \frac{\log \left(\frac{x + 1}{x}\right)}{n}\right) - 0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array}\]

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 1.280852323654058e-211)
   (+
    (* -0.5 (/ (pow (log x) 2.0) (* n n)))
    (-
     (+
      (+
       (* 0.5 (/ (pow (log (+ x 1.0)) 2.0) (* n n)))
       (* 0.16666666666666666 (pow (/ (log (+ x 1.0)) n) 3.0)))
      (/ (- (log (+ x 1.0)) (log x)) n))
     (* 0.16666666666666666 (pow (/ (log x) n) 3.0))))
   (if (<= x 2.2097631365031477e-202)
     (*
      (sqrt (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))
      (sqrt (exp (log (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))))
     (if (<= x 147972.98971215458)
       (+
        (* -0.5 (/ (pow (log x) 2.0) (* n n)))
        (-
         (+
          (+
           (* 0.5 (/ (pow (log (+ x 1.0)) 2.0) (* n n)))
           (* 0.16666666666666666 (pow (/ (log (+ x 1.0)) n) 3.0)))
          (/ (log (/ (+ x 1.0) x)) n))
         (* 0.16666666666666666 (pow (/ (log x) n) 3.0))))
       (/ (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 <= 1.280852323654058e-211) {
		tmp = (-0.5 * (pow(log(x), 2.0) / (n * n))) + ((((0.5 * (pow(log(x + 1.0), 2.0) / (n * n))) + (0.16666666666666666 * pow((log(x + 1.0) / n), 3.0))) + ((log(x + 1.0) - log(x)) / n)) - (0.16666666666666666 * pow((log(x) / n), 3.0)));
	} else if (x <= 2.2097631365031477e-202) {
		tmp = sqrt(pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n))) * sqrt(exp(log(pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n)))));
	} else if (x <= 147972.98971215458) {
		tmp = (-0.5 * (pow(log(x), 2.0) / (n * n))) + ((((0.5 * (pow(log(x + 1.0), 2.0) / (n * n))) + (0.16666666666666666 * pow((log(x + 1.0) / n), 3.0))) + (log((x + 1.0) / x) / n)) - (0.16666666666666666 * pow((log(x) / n), 3.0)));
	} else {
		tmp = exp(log(x) / n) / (x * n);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus n

Alternatives

Alternative 1
Error	58.6
Cost	421120

\[\frac{\left(n \cdot n\right) \cdot \left({n}^{3} \cdot \left({\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3}\right)}^{3} + {\left(\frac{\log \left(x + 1\right) - \log x}{n}\right)}^{3}\right) - \left(0.16666666666666666 \cdot {\log x}^{3}\right) \cdot \left(\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3}\right) \cdot \left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3}\right) + \frac{\log \left(x + 1\right) - \log x}{n} \cdot \left(\frac{\log \left(x + 1\right) - \log x}{n} + \left({\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3} \cdot -0.16666666666666666 + \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} \cdot -0.5\right)\right)\right)\right) + \left(\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3}\right) \cdot \left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3}\right) + \frac{\log \left(x + 1\right) - \log x}{n} \cdot \left(\frac{\log \left(x + 1\right) - \log x}{n} + \left({\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3} \cdot -0.16666666666666666 + \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} \cdot -0.5\right)\right)\right) \cdot \left({n}^{3} \cdot \left(-0.5 \cdot {\log x}^{2}\right)\right)}{\left(\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3}\right) \cdot \left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3}\right) + \frac{\log \left(x + 1\right) - \log x}{n} \cdot \left(\frac{\log \left(x + 1\right) - \log x}{n} + \left({\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3} \cdot -0.16666666666666666 + \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} \cdot -0.5\right)\right)\right) \cdot {n}^{5}}\]

Alternative 2
Error	55.0
Cost	374080

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

Alternative 3
Error	55.5
Cost	307264

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

Alternative 4
Error	50.9
Cost	214080

\[\frac{\left(n \cdot n\right) \cdot \left(\left(\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3}\right) + \frac{\log \left(x + 1\right) - \log x}{n}\right) \cdot \left(\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3}\right) + \frac{\log \left(x + 1\right) - \log x}{n}\right) + -0.027777777777777776 \cdot {\left(\frac{\log x}{n}\right)}^{6}\right) + \left(-0.5 \cdot {\log x}^{2}\right) \cdot \left(\left(\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3}\right) + \frac{\log \left(x + 1\right) - \log x}{n}\right) + 0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right)}{\left(n \cdot n\right) \cdot \left(\left(\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3}\right) + \frac{\log \left(x + 1\right) - \log x}{n}\right) + 0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right)}\]

Alternative 5
Error	30.9
Cost	192832

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

Alternative 6
Error	38.7
Cost	146752

\[\left(\left(\left(\frac{e^{\frac{\log x}{n}}}{n \cdot x} + \frac{e^{\frac{\log x}{n}}}{{x}^{3}} \cdot \left(\frac{0.16666666666666666}{{n}^{3}} + \frac{0.3333333333333333}{n}\right)\right) + 0.041666666666666664 \cdot \frac{e^{\frac{\log x}{n}}}{{x}^{4} \cdot {n}^{4}}\right) + \frac{e^{\frac{\log x}{n}}}{n \cdot n} \cdot \left(\frac{0.5}{x \cdot x} + \frac{0.4583333333333333}{{x}^{4}}\right)\right) - \left(\frac{e^{\frac{\log x}{n}}}{{x}^{4}} \cdot \left(\frac{0.25}{n} + \frac{0.25}{{n}^{3}}\right) + 0.5 \cdot \left(\frac{e^{\frac{\log x}{n}}}{x \cdot \left(n \cdot x\right)} + \frac{e^{\frac{\log x}{n}}}{n \cdot \left(n \cdot {x}^{3}\right)}\right)\right)\]

Alternative 7
Error	41.8
Cost	132992

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

Alternative 8
Error	28.5
Cost	112320

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

Alternative 9
Error	28.2
Cost	92736

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

Alternative 10
Error	35.3
Cost	86208

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

Alternative 11
Error	45.5
Cost	79616

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

Alternative 12
Error	28.2
Cost	79616

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

Alternative 13
Error	28.4
Cost	79552

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

Alternative 14
Error	37.4
Cost	73856

\[\left(\left(\frac{e^{\frac{\log x}{n}}}{n \cdot x} + \frac{e^{\frac{\log x}{n}}}{{x}^{3}} \cdot \left(\frac{0.16666666666666666}{{n}^{3}} + \frac{0.3333333333333333}{n}\right)\right) + \frac{e^{\frac{\log x}{n}}}{x \cdot x} \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)\right) + -0.5 \cdot \frac{e^{\frac{\log x}{n}}}{n \cdot \left(n \cdot {x}^{3}\right)}\]

Alternative 15
Error	28.2
Cost	73216

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

Alternative 16
Error	46.1
Cost	67648

\[-0.5 \cdot \frac{{\log x}^{2}}{n \cdot n} + \left(\left(\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3}\right) + \frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\frac{0.5}{x \cdot x} + \frac{0.25}{{x}^{4}}\right)}{n}\right) - 0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right)\]

Alternative 17
Error	28.1
Cost	66752

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

Alternative 18
Error	28.0
Cost	60352

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

Alternative 19
Error	45.1
Cost	53824

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

Alternative 20
Error	38.2
Cost	53376

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

Alternative 21
Error	33.3
Cost	52672

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

Alternative 22
Error	32.9
Cost	52672

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

Alternative 23
Error	17.5
Cost	40000

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

Alternative 24
Error	32.9
Cost	39872

\[\left({x}^{\left(\frac{0.5}{n}\right)} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{0.5}{n}\right)}\right)\]

Alternative 25
Error	33.3
Cost	39872

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

Alternative 26
Error	39.7
Cost	33216

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

Alternative 27
Error	35.8
Cost	27200

\[\frac{e^{\frac{\log x}{n}}}{n \cdot x} + \frac{e^{\frac{\log x}{n}}}{x \cdot x} \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)\]

Alternative 28
Error	44.5
Cost	27072

\[\frac{{\left(x + 1\right)}^{\left(\frac{2}{n}\right)} - {x}^{\left(\frac{2}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\]

Alternative 29
Error	40.5
Cost	26816

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

Alternative 30
Error	33.3
Cost	26368

\[{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{2}\]

Alternative 31
Error	33.3
Cost	26304

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

Alternative 32
Error	32.9
Cost	26304

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

Alternative 33
Error	32.8
Cost	13504

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

Alternative 34
Error	55.0
Cost	13376

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

Alternative 35
Error	17.2
Cost	13248

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

Alternative 36
Error	29.1
Cost	13248

\[\frac{e^{\frac{\log x}{n}}}{n \cdot x}\]

Alternative 37
Error	44.3
Cost	13120

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

Alternative 38
Error	39.0
Cost	64

\[0\]

Alternative 39
Error	56.4
Cost	64

\[1\]

Alternative 40
Error	61.9
Cost	64

\[-1\]

Derivation

1. Split input into 4 regimes

2. if x < 1.2808523236540581e-211

   1. Initial program 41.8

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

   2. Taylor expanded around inf 19.3

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

   3. Simplified 19.3

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

   4. Simplified 19.3

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

   if 1.2808523236540581e-211 < x < 2.20976313650314774e-202

   1. Initial program 41.2

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt_binary64_100 42.2

      \[\leadsto \color{blue}{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\]
   4. Using strategy rm

   5. Applied add-exp-log_binary64_116 42.2

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

   6. Simplified 42.2

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

   if 2.20976313650314774e-202 < x < 147972.98971215458

   1. Initial program 50.7

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

   2. Taylor expanded around inf 10.0

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

   3. Simplified 10.0

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

   5. Applied diff-log_binary64_170 10.0

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

   6. Simplified 10.0

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

   if 147972.98971215458 < x

   1. Initial program 19.9

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

   2. Taylor expanded around inf 1.2

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

   3. Simplified 1.2

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

   4. Simplified 1.2

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{n \cdot x}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.280852323654058 \cdot 10^{-211}:\\ \;\;\;\;-0.5 \cdot \frac{{\log x}^{2}}{n \cdot n} + \left(\left(\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3}\right) + \frac{\log \left(x + 1\right) - \log x}{n}\right) - 0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right)\\ \mathbf{elif}\;x \leq 2.2097631365031477 \cdot 10^{-202}:\\ \;\;\;\;\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\\ \mathbf{elif}\;x \leq 147972.98971215458:\\ \;\;\;\;-0.5 \cdot \frac{{\log x}^{2}}{n \cdot n} + \left(\left(\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3}\right) + \frac{\log \left(\frac{x + 1}{x}\right)}{n}\right) - 0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2021022 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))