Average Error: 29.1 → 22.0
Time: 28.3s
Precision: 64
Internal Precision: 128
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.0024609209182995073:\\ \;\;\;\;\left(\log \left(\sqrt[3]{e^{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}}}\right) + \log \left(\sqrt[3]{e^{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}}} \cdot \sqrt[3]{e^{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}}}\right)\right) \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 3.3536702913919863 \cdot 10^{-07}:\\ \;\;\;\;\left(\frac{\frac{-1}{2}}{x} + 1\right) \cdot \frac{1}{x \cdot n} + \frac{\log x}{n \cdot \left(x \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (/ 1 n) < -0.0024609209182995073

    1. Initial program 0.3

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

    3. Applied add-log-exp0.5

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

    4. Applied add-log-exp0.5

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

    5. Applied diff-log0.5

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

    6. Simplified0.5

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

    8. Applied add-sqr-sqrt0.7

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

    9. Applied exp-prod0.7

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

    10. Applied log-pow0.7

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

    12. Applied add-log-exp0.7

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

    14. Applied add-cube-cbrt0.7

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

    15. Applied log-prod0.7

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

    if -0.0024609209182995073 < (/ 1 n) < 3.3536702913919863e-07

    1. Initial program 44.9

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

    2. Taylor expanded around inf 32.4

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

    3. Simplified32.4

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

    if 3.3536702913919863e-07 < (/ 1 n)

    1. Initial program 22.0

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

    3. Applied add-log-exp22.0

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

    4. Applied add-log-exp22.0

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

    5. Applied diff-log22.0

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

    6. Simplified22.0

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

    8. Applied add-sqr-sqrt22.0

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

    9. Applied exp-prod22.0

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

    10. Applied log-pow22.0

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

    12. Applied add-log-exp22.0

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

    14. Applied rem-log-exp22.0

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

  4. Final simplification22.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.0024609209182995073:\\ \;\;\;\;\left(\log \left(\sqrt[3]{e^{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}}}\right) + \log \left(\sqrt[3]{e^{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}}} \cdot \sqrt[3]{e^{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}}}\right)\right) \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 3.3536702913919863 \cdot 10^{-07}:\\ \;\;\;\;\left(\frac{\frac{-1}{2}}{x} + 1\right) \cdot \frac{1}{x \cdot n} + \frac{\log x}{n \cdot \left(x \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))