2nthrt (problem 3.4.6)

Average Error: 31.9 → 22.8

Time: 16.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -177055548.10463205 \lor \neg \left(n \le 5036922.0148078054\right):\\ \;\;\;\;\left(\frac{1}{x} \cdot \left(\frac{1}{n} + 0\right) + \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left(\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}}\right)\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 2 regimes

2. if n < -177055548.10463205 or 5036922.0148078054 < n

   1. Initial program 45.3

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

   2. Taylor expanded around inf 32.6

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

   3. Simplified 32.0

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

   if -177055548.10463205 < n < 5036922.0148078054

   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-sqr-sqrt 2.6

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

   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 2.6

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

   7. Applied difference-of-squares 2.6

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

   9. Applied add-cube-cbrt 2.6

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

   11. Applied add-cube-cbrt 2.6

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

4. Final simplification 22.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -177055548.10463205 \lor \neg \left(n \le 5036922.0148078054\right):\\ \;\;\;\;\left(\frac{1}{x} \cdot \left(\frac{1}{n} + 0\right) + \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left(\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020157 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))