Average Error: 33.5 → 23.3
Time: 1.4m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.7397607380483905 \cdot 10^{-09}:\\ \;\;\;\;\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + {\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.3035418001463924 \cdot 10^{-13}:\\ \;\;\;\;\left(\frac{\log x}{n} - \frac{\frac{1}{2}}{x}\right) \cdot \left({\left(\frac{1}{x}\right)}^{\left(3 \cdot \frac{1}{3}\right)} \cdot {\left(\frac{1}{n}\right)}^{\left(3 \cdot \frac{1}{3}\right)}\right) + {\left(\frac{1}{x}\right)}^{\left(3 \cdot \frac{1}{3}\right)} \cdot {\left(\frac{1}{n}\right)}^{\left(3 \cdot \frac{1}{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (/ 1 n) < -3.7397607380483905e-09

    1. Initial program 1.4

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

    3. Applied add-sqr-sqrt1.5

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

    4. Applied add-sqr-sqrt1.6

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

    5. Applied unpow-prod-down1.5

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

    6. Applied difference-of-squares1.5

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

    if -3.7397607380483905e-09 < (/ 1 n) < 2.3035418001463924e-13

    1. Initial program 45.7

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

    3. Applied add-cbrt-cube45.7

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

    4. Taylor expanded around inf 48.3

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

    5. Simplified31.8

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

    if 2.3035418001463924e-13 < (/ 1 n)

    1. Initial program 8.3

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

    3. Applied add-exp-log8.3

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

    4. Applied pow-exp8.3

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

    5. Simplified5.5

      \[\leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification23.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.7397607380483905 \cdot 10^{-09}:\\ \;\;\;\;\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + {\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.3035418001463924 \cdot 10^{-13}:\\ \;\;\;\;\left(\frac{\log x}{n} - \frac{\frac{1}{2}}{x}\right) \cdot \left({\left(\frac{1}{x}\right)}^{\left(3 \cdot \frac{1}{3}\right)} \cdot {\left(\frac{1}{n}\right)}^{\left(3 \cdot \frac{1}{3}\right)}\right) + {\left(\frac{1}{x}\right)}^{\left(3 \cdot \frac{1}{3}\right)} \cdot {\left(\frac{1}{n}\right)}^{\left(3 \cdot \frac{1}{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018219 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))