2nthrt (problem 3.4.6)

Average Error: 33.0 → 23.8

Time: 14.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.24462942643905788 \cdot 10^{-12}:\\ \;\;\;\;\left(\log \left(e^{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)}}\right) + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \le 2.75596332523924502 \cdot 10^{-10}:\\ \;\;\;\;\left(\frac{1}{x} \cdot \left(\frac{1}{n} + 0\right) + \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left({\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}\right)\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -6.24462942643905788e-12

   1. Initial program 1.8

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

   3. Applied add-sqr-sqrt 1.9

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

   4. Applied add-sqr-sqrt 1.9

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

   5. Applied unpow-prod-down 1.8

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

   6. Applied difference-of-squares 1.9

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

   8. Applied add-cbrt-cube 2.0

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

   9. Simplified 2.0

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

   11. Applied add-log-exp 2.0

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

   if -6.24462942643905788e-12 < (/ 1.0 n) < 2.75596332523924502e-10

   1. Initial program 45.4

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

   2. Taylor expanded around inf 32.8

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

   3. Simplified 32.2

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

   if 2.75596332523924502e-10 < (/ 1.0 n)

   1. Initial program 7.0

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

   3. Applied add-sqr-sqrt 7.1

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

   4. Applied add-sqr-sqrt 7.1

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

   5. Applied unpow-prod-down 7.1

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

   6. Applied difference-of-squares 7.1

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

   8. Applied add-sqr-sqrt 7.1

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

   9. Applied unpow-prod-down 7.1

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

   10. Applied sqrt-prod 7.1

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

   11. Applied add-sqr-sqrt 7.1

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

   12. Applied sqrt-prod 7.1

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

   13. Applied unpow-prod-down 7.1

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

   14. Applied difference-of-squares 7.1

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

   16. Applied add-sqr-sqrt 7.1

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

   17. Applied sqrt-prod 7.1

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

   18. Applied unpow-prod-down 7.1

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

4. Final simplification 23.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.24462942643905788 \cdot 10^{-12}:\\ \;\;\;\;\left(\log \left(e^{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)}}\right) + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \le 2.75596332523924502 \cdot 10^{-10}:\\ \;\;\;\;\left(\frac{1}{x} \cdot \left(\frac{1}{n} + 0\right) + \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left({\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))