Average Error: 32.9 → 24.0
Time: 56.2s
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -1771305.9941476847:\\ \;\;\;\;\left(\frac{\frac{\log x}{n \cdot n}}{x} + \frac{1}{x \cdot n}\right) - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\\ \mathbf{elif}\;n \le 7.810905648762942 \cdot 10^{+22}:\\ \;\;\;\;{\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{x \cdot n}}{x} - \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if n < -1771305.9941476847

    1. Initial program 45.4

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

    2. Initial simplification45.4

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

    3. Taylor expanded around -inf 63.2

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

    4. Simplified32.4

      \[\leadsto \color{blue}{\left(\left(\frac{1}{x \cdot n} + 0\right) + \frac{\frac{\log x}{n \cdot n}}{x}\right) - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}}\]

    if -1771305.9941476847 < n < 7.810905648762942e+22

    1. Initial program 6.3

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

    2. Initial simplification6.3

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

    4. Applied add-cube-cbrt6.4

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

    5. Applied unpow-prod-down6.4

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

    if 7.810905648762942e+22 < n

    1. Initial program 44.7

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

    2. Initial simplification44.7

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

    3. Taylor expanded around inf 31.8

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

    4. Simplified31.8

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

  4. Final simplification24.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1771305.9941476847:\\ \;\;\;\;\left(\frac{\frac{\log x}{n \cdot n}}{x} + \frac{1}{x \cdot n}\right) - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\\ \mathbf{elif}\;n \le 7.810905648762942 \cdot 10^{+22}:\\ \;\;\;\;{\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{x \cdot n}}{x} - \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\\ \end{array}\]

Runtime

Time bar (total: 56.2s)Debug logProfile

herbie shell --seed 2018234 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))