2nthrt (problem 3.4.6)

Average Error: 32.1 → 23.5

Time: 20.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2508.27711592578044:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \left(2 \cdot \log \left(\sqrt[3]{e^{{x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt[3]{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 8.9845027404309597 \cdot 10^{-15}:\\ \;\;\;\;\left(\left(\frac{1}{x} \cdot \frac{0}{{n}^{2}} - 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{\left({\left(\sqrt[3]{n}\right)}^{4} \cdot x\right) \cdot {\left(\sqrt[3]{n}\right)}^{2}}\right) - 0.5 \cdot \frac{1}{{x}^{2} \cdot n}\right) + \frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + \left(-{x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -2508.27711592578044

   1. Initial program 0

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

   3. Applied add-log-exp 0

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

   5. Applied add-cube-cbrt 0

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

   6. Applied log-prod 0

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

   7. Simplified 0

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)} + \log \left(\sqrt[3]{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)\right)\]

   if -2508.27711592578044 < (/ 1.0 n) < 8.9845027404309597e-15

   1. Initial program 44.4

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

   3. Applied add-log-exp 44.5

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

   4. Taylor expanded around inf 32.7

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

   5. Simplified 32.2

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

   7. Applied add-cube-cbrt 32.2

      \[\leadsto \left(\left(\frac{1}{x} \cdot \frac{0}{{n}^{2}} - 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {\color{blue}{\left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}\right)}}^{2}}\right) - 0.5 \cdot \frac{1}{{x}^{2} \cdot n}\right) + \frac{\frac{1}{n}}{x}\]

   8. Applied unpow-prod-down 32.2

      \[\leadsto \left(\left(\frac{1}{x} \cdot \frac{0}{{n}^{2}} - 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot \color{blue}{\left({\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)}^{2} \cdot {\left(\sqrt[3]{n}\right)}^{2}\right)}}\right) - 0.5 \cdot \frac{1}{{x}^{2} \cdot n}\right) + \frac{\frac{1}{n}}{x}\]

   9. Applied associate-*r* 32.2

      \[\leadsto \left(\left(\frac{1}{x} \cdot \frac{0}{{n}^{2}} - 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{\color{blue}{\left(x \cdot {\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)}^{2}\right) \cdot {\left(\sqrt[3]{n}\right)}^{2}}}\right) - 0.5 \cdot \frac{1}{{x}^{2} \cdot n}\right) + \frac{\frac{1}{n}}{x}\]

   10. Simplified 32.2

       \[\leadsto \left(\left(\frac{1}{x} \cdot \frac{0}{{n}^{2}} - 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{\color{blue}{\left({\left(\sqrt[3]{n}\right)}^{4} \cdot x\right)} \cdot {\left(\sqrt[3]{n}\right)}^{2}}\right) - 0.5 \cdot \frac{1}{{x}^{2} \cdot n}\right) + \frac{\frac{1}{n}}{x}\]

   if 8.9845027404309597e-15 < (/ 1.0 n)

   1. Initial program 7.5

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

   3. Applied flip-- 7.5

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

   4. Simplified 7.5

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

4. Final simplification 23.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2508.27711592578044:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \left(2 \cdot \log \left(\sqrt[3]{e^{{x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt[3]{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 8.9845027404309597 \cdot 10^{-15}:\\ \;\;\;\;\left(\left(\frac{1}{x} \cdot \frac{0}{{n}^{2}} - 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{\left({\left(\sqrt[3]{n}\right)}^{4} \cdot x\right) \cdot {\left(\sqrt[3]{n}\right)}^{2}}\right) - 0.5 \cdot \frac{1}{{x}^{2} \cdot n}\right) + \frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + \left(-{x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))