2nthrt (problem 3.4.6)

Average Error: 33.1 → 24.3

Time: 21.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -41.861711160932543 \lor \neg \left(n \le 914165034.27439833\right):\\ \;\;\;\;\left(\frac{1}{x} \cdot \left(\frac{1}{n} + 0\right) + \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(e^{\sqrt[3]{\log \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \log \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)}}\right)}^{\left(\sqrt[3]{\sqrt[3]{{\left(\log \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)\right)}^{3}}}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 2 regimes

2. if n < -41.861711160932543 or 914165034.27439833 < n

   1. Initial program 45.0

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

   2. Taylor expanded around inf 33.2

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

   3. Simplified 32.6

      \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\frac{1}{n} + 0\right) + \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}}\]

   if -41.861711160932543 < n < 914165034.27439833

   1. Initial program 2.7

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

   3. Applied add-sqr-sqrt 2.7

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

   4. Applied unpow-prod-down 2.7

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

   5. Applied add-sqr-sqrt 2.7

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

   6. Applied unpow-prod-down 2.7

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

   7. Applied difference-of-squares 2.7

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

   9. Applied add-exp-log 3.2

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

   11. Applied add-cbrt-cube 3.2

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

   12. Simplified 3.2

       \[\leadsto \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot e^{\sqrt[3]{\color{blue}{{\left(\log \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)\right)}^{3}}}}\]
   13. Using strategy rm

   14. Applied add-cube-cbrt 3.2

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

   15. Applied cbrt-prod 3.2

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

   16. Applied exp-prod 3.2

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

   17. Simplified 3.2

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

4. Final simplification 24.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -41.861711160932543 \lor \neg \left(n \le 914165034.27439833\right):\\ \;\;\;\;\left(\frac{1}{x} \cdot \left(\frac{1}{n} + 0\right) + \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(e^{\sqrt[3]{\log \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \log \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)}}\right)}^{\left(\sqrt[3]{\sqrt[3]{{\left(\log \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)\right)}^{3}}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))