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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*} \cdot \left(\left(\sqrt[3]{\sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*}} \cdot \sqrt[3]{\sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*}}\right) \cdot \sqrt[3]{\sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*}}\right)\right) \cdot \sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*} \le -3560.253012610007:\\ \;\;\;\;{e}^{\left(\frac{\log_* (1 + x)}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\left(\sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*} \cdot \left(\left(\sqrt[3]{\sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*}} \cdot \sqrt[3]{\sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*}}\right) \cdot \sqrt[3]{\sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*}}\right)\right) \cdot \sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*} \le -8.631719590496053 \cdot 10^{-307}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\log x \cdot \frac{1}{n}\right))_*\\ \mathbf{if}\;\left(\sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*} \cdot \left(\left(\sqrt[3]{\sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*}} \cdot \sqrt[3]{\sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*}}\right) \cdot \sqrt[3]{\sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*}}\right)\right) \cdot \sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*} \le 0.0:\\ \;\;\;\;\frac{\frac{\log x}{n \cdot n}}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{1}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{1}{\frac{n}{\log x}}\right))_*\\ \end{array}\]

Derivation

Split input into 4 regimes
if (* (* (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n)))) (* (* (cbrt (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n))))) (cbrt (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n)))))) (cbrt (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n))))))) (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n))))) < -3560.253012610007
1. Initial program 1.2
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Using strategy rm
3. Applied add-exp-log1.2
  \[\leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
4. Applied pow-exp1.2
  \[\leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
5. Applied simplify0.1
  \[\leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
6. Using strategy rm
7. Applied *-un-lft-identity0.1
  \[\leadsto e^{\color{blue}{1 \cdot \frac{\log_* (1 + x)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
8. Applied exp-prod0.1
  \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\frac{\log_* (1 + x)}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\]
9. Applied simplify0.1
  \[\leadsto {\color{blue}{e}}^{\left(\frac{\log_* (1 + x)}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
if -3560.253012610007 < (* (* (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n)))) (* (* (cbrt (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n))))) (cbrt (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n)))))) (cbrt (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n))))))) (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n))))) < -8.631719590496053e-307
1. Initial program 58.2
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Using strategy rm
3. Applied add-exp-log58.3
  \[\leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
4. Applied pow-exp58.3
  \[\leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
5. Applied simplify58.3
  \[\leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
6. Taylor expanded around inf 59.2
  \[\leadsto e^{\frac{\log_* (1 + x)}{n}} - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{{\left(\log \left(\frac{1}{x}\right)\right)}^{2}}{{n}^{2}} + 1\right) - \frac{\log \left(\frac{1}{x}\right)}{n}\right)}\]
7. Applied simplify3.8
  \[\leadsto \color{blue}{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*}\]
8. Using strategy rm
9. Applied div-inv4.0
  \[\leadsto (e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \color{blue}{\left(\log x \cdot \frac{1}{n}\right)})_*\]
if -8.631719590496053e-307 < (* (* (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n)))) (* (* (cbrt (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n))))) (cbrt (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n)))))) (cbrt (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n))))))) (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n))))) < 0.0
1. Initial program 28.6
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Taylor expanded around inf 1.4
  \[\leadsto \color{blue}{\frac{1}{n \cdot x} - \left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^{2}} + \frac{\log \left(\frac{1}{x}\right)}{{n}^{2} \cdot x}\right)}\]
3. Applied simplify1.4
  \[\leadsto \color{blue}{\frac{\frac{\log x}{n \cdot n}}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{1}{x \cdot n}\right)}\]
if 0.0 < (* (* (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n)))) (* (* (cbrt (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n))))) (cbrt (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n)))))) (cbrt (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n))))))) (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n)))))
1. Initial program 58.1
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Using strategy rm
3. Applied add-exp-log58.1
  \[\leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
4. Applied pow-exp58.1
  \[\leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
5. Applied simplify58.1
  \[\leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
6. Taylor expanded around inf 59.2
  \[\leadsto e^{\frac{\log_* (1 + x)}{n}} - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{{\left(\log \left(\frac{1}{x}\right)\right)}^{2}}{{n}^{2}} + 1\right) - \frac{\log \left(\frac{1}{x}\right)}{n}\right)}\]
7. Applied simplify4.6
  \[\leadsto \color{blue}{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*}\]
8. Using strategy rm
9. Applied clear-num4.7
  \[\leadsto (e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \color{blue}{\left(\frac{1}{\frac{n}{\log x}}\right)})_*\]
Recombined 4 regimes into one program.

Error

Derivation

Runtime