2nthrt (problem 3.4.6)

Average Error: 32.8 → 24.3

Time: 13.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.011482869746324429:\\ \;\;\;\;\left(\sqrt[3]{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left({x}^{\left(\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}}\right)}\right)}^{\left(\frac{1}{\sqrt[3]{n}}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 7.0917597959631381 \cdot 10^{-40}:\\ \;\;\;\;\left(\frac{1}{x} \cdot \left(\frac{1}{n} + 0\right) + \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + \left(-{x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -0.011482869746324429

   1. Initial program 0.2

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

   3. Applied add-cube-cbrt 0.2

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

   5. Applied add-cube-cbrt 0.2

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

   6. Applied *-un-lft-identity 0.2

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

   7. Applied times-frac 0.2

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

   8. Applied pow-unpow 0.2

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

   10. Applied add-sqr-sqrt 0.2

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

   11. Applied unpow-prod-down 0.2

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

   12. Applied add-sqr-sqrt 0.2

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

   13. Applied unpow-prod-down 0.2

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

   14. Applied difference-of-squares 0.2

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

   16. Applied add-sqr-sqrt 0.2

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

   17. Applied unpow-prod-down 0.2

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

   18. Applied add-sqr-sqrt 0.2

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

   19. Applied unpow-prod-down 0.2

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

   20. Applied difference-of-squares 0.2

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

   if -0.011482869746324429 < (/ 1.0 n) < 7.0917597959631381e-40

   1. Initial program 44.3

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

   2. Taylor expanded around inf 32.4

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

   3. Simplified 31.8

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

   if 7.0917597959631381e-40 < (/ 1.0 n)

   1. Initial program 19.6

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

   3. Applied flip-- 19.6

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

   4. Simplified 19.6

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

4. Final simplification 24.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.011482869746324429:\\ \;\;\;\;\left(\sqrt[3]{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left({x}^{\left(\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}}\right)}\right)}^{\left(\frac{1}{\sqrt[3]{n}}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 7.0917597959631381 \cdot 10^{-40}:\\ \;\;\;\;\left(\frac{1}{x} \cdot \left(\frac{1}{n} + 0\right) + \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + \left(-{x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))