2nthrt (problem 3.4.6)

Average Error: 32.9 → 24.5

Time: 11.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.34903097097955769 \cdot 10^{-7}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \left({\left(\sqrt{{\left(\sqrt{\sqrt{x}}\right)}^{\left(\frac{1}{n}\right)}}\right)}^{3} \cdot {\left(\sqrt{{\left(\sqrt{\sqrt{x}}\right)}^{\left(\frac{1}{n}\right)}}\right)}^{3}\right) \cdot \sqrt{e^{\log \left(\sqrt{x}\right) \cdot \frac{1}{n}}}\\ \mathbf{elif}\;\frac{1}{n} \le 7.2869097479865114 \cdot 10^{-30}:\\ \;\;\;\;\left(\frac{1}{x} \cdot \left(\frac{1}{n} + 0\right) + \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + \left(-{x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -1.34903097097955769e-7

   1. Initial program 1.3

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

   3. Applied add-sqr-sqrt 1.4

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

   4. Applied unpow-prod-down 1.4

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

   6. Applied add-exp-log 1.5

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

   7. Applied pow-exp 1.5

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

   9. Applied add-sqr-sqrt 1.5

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

   10. Applied associate-*r* 1.5

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

   11. Simplified 1.5

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

   13. Applied add-sqr-sqrt 1.5

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

   14. Applied sqrt-prod 1.5

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

   15. Applied unpow-prod-down 1.5

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

   16. Applied sqrt-prod 1.5

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

   17. Applied unpow-prod-down 1.5

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

   if -1.34903097097955769e-7 < (/ 1.0 n) < 7.2869097479865114e-30

   1. Initial program 45.1

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

   2. Taylor expanded around inf 33.4

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

   3. Simplified 32.8

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

   if 7.2869097479865114e-30 < (/ 1.0 n)

   1. Initial program 14.5

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

   3. Applied flip-- 14.5

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

   4. Simplified 14.4

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

4. Final simplification 24.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.34903097097955769 \cdot 10^{-7}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \left({\left(\sqrt{{\left(\sqrt{\sqrt{x}}\right)}^{\left(\frac{1}{n}\right)}}\right)}^{3} \cdot {\left(\sqrt{{\left(\sqrt{\sqrt{x}}\right)}^{\left(\frac{1}{n}\right)}}\right)}^{3}\right) \cdot \sqrt{e^{\log \left(\sqrt{x}\right) \cdot \frac{1}{n}}}\\ \mathbf{elif}\;\frac{1}{n} \le 7.2869097479865114 \cdot 10^{-30}:\\ \;\;\;\;\left(\frac{1}{x} \cdot \left(\frac{1}{n} + 0\right) + \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + \left(-{x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

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))))