2nthrt (problem 3.4.6)

Average Error: 32.9 → 23.8

Time: 14.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.35971998423245161 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \left(\left(\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt[3]{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 3.9907891505810576 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{1}{x} \cdot \left(\frac{1}{n} + 0\right) + \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + \left(-{x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -1.35971998423245161e-22

   1. Initial program 5.5

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

   3. Applied add-sqr-sqrt 5.5

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

   4. Applied unpow-prod-down 5.6

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

   5. Applied add-sqr-sqrt 5.6

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

   6. Applied unpow-prod-down 5.6

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

   7. Applied difference-of-squares 5.6

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

   9. Applied add-sqr-sqrt 5.6

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

   10. Applied associate-*l* 5.6

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

   12. Applied add-cube-cbrt 5.6

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

   14. Applied add-cube-cbrt 5.6

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

   15. Applied unpow-prod-down 5.6

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

   if -1.35971998423245161e-22 < (/ 1.0 n) < 3.9907891505810576e-5

   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 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 3.9907891505810576e-5 < (/ 1.0 n)

   1. Initial program 5.4

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

   3. Applied flip-- 5.4

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

   4. Simplified 5.4

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

4. Final simplification 23.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.35971998423245161 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \left(\left(\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt[3]{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 3.9907891505810576 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{1}{x} \cdot \left(\frac{1}{n} + 0\right) + \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + \left(-{x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))