2nthrt (problem 3.4.6)

Average Error: 32.9 → 24.1

Time: 13.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -145.036925634951245:\\ \;\;\;\;\left(\frac{1}{x} \cdot \left(\frac{1}{n} + 0\right) + \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{elif}\;n \le 390855483.729558289:\\ \;\;\;\;{\left(e^{\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}^{\left(\sqrt[3]{\log \left(\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(\left(\left(-\left(\log n + \log x\right)\right) + 1 \cdot \frac{\log 1}{n}\right) + \log 1\right) - 0.5 \cdot \frac{1}{x}\right) - 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if n < -145.036925634951245

   1. Initial program 44.8

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

   2. Taylor expanded around inf 33.0

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

   3. Simplified 32.5

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

   if -145.036925634951245 < n < 390855483.729558289

   1. Initial program 2.6

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

   3. Applied add-exp-log 3.1

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

   5. Applied add-cube-cbrt 3.1

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

   6. Applied exp-prod 3.1

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

   8. Applied add-sqr-sqrt 3.2

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

   9. Applied unpow-prod-down 3.2

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

   10. Applied add-sqr-sqrt 3.1

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

   11. Applied unpow-prod-down 3.1

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

   12. Applied difference-of-squares 3.1

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

   if 390855483.729558289 < n

   1. Initial program 45.1

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

   3. Applied add-exp-log 45.1

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

   4. Taylor expanded around inf 32.3

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

   5. Simplified 32.3

      \[\leadsto e^{\color{blue}{\left(\left(\left(\left(-\left(\log n + \log x\right)\right) + 1 \cdot \frac{\log 1}{n}\right) + \log 1\right) - 0.5 \cdot \frac{1}{x}\right) - 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 24.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -145.036925634951245:\\ \;\;\;\;\left(\frac{1}{x} \cdot \left(\frac{1}{n} + 0\right) + \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{elif}\;n \le 390855483.729558289:\\ \;\;\;\;{\left(e^{\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}^{\left(\sqrt[3]{\log \left(\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(\left(\left(-\left(\log n + \log x\right)\right) + 1 \cdot \frac{\log 1}{n}\right) + \log 1\right) - 0.5 \cdot \frac{1}{x}\right) - 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020171 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))