Average Error: 33.3 → 23.9
Time: 1.1m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -7.103038796341682 \cdot 10^{-23}:\\ \;\;\;\;(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) + \left(-{x}^{\left(\frac{1}{n}\right)}\right))_*\\ \mathbf{elif}\;\frac{1}{n} \le 5.793954106697829 \cdot 10^{-10}:\\ \;\;\;\;(\left(\frac{\frac{1}{4}}{x}\right) \cdot \left(\frac{\log x}{n \cdot n}\right) + \left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{x}\right) \cdot \frac{\frac{1}{x}}{n}\right))_* \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{e^{\frac{\log_* (1 + x)}{n}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{\frac{\log_* (1 + x)}{n}}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{\left({x}^{\left(\frac{1}{n}\right)} \cdot e^{\frac{\log_* (1 + x)}{n}} + {x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right) + e^{\frac{\log_* (1 + x)}{n}} \cdot e^{\frac{\log_* (1 + x)}{n}}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (/ 1 n) < -7.103038796341682e-23

    1. Initial program 5.3

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

    3. Applied add-cube-cbrt5.4

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

    4. Applied fma-neg5.5

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

    if -7.103038796341682e-23 < (/ 1 n) < 5.793954106697829e-10

    1. Initial program 45.4

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

    3. Applied add-exp-log45.4

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

    4. Applied pow-exp45.4

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

    5. Simplified45.4

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

    7. Applied add-cbrt-cube45.5

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

    9. Applied add-sqr-sqrt45.5

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

    10. Applied add-sqr-sqrt45.5

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

    11. Applied difference-of-squares45.5

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

    12. Simplified45.5

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

    13. Simplified45.4

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

    14. Taylor expanded around inf 33.0

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

    15. Simplified32.3

      \[\leadsto \left(\sqrt{e^{\frac{\log_* (1 + x)}{n}}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \color{blue}{(\left(\frac{\frac{1}{4}}{x}\right) \cdot \left(\frac{\log x}{n \cdot n}\right) + \left(\frac{\frac{1}{x}}{n} \cdot \left(\frac{1}{2} - \frac{\frac{1}{4}}{x}\right)\right))_*}\]

    if 5.793954106697829e-10 < (/ 1 n)

    1. Initial program 6.9

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

    3. Applied add-exp-log7.0

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

    4. Applied pow-exp7.0

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

    5. Simplified3.7

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

    7. Applied flip3--3.8

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

  4. Final simplification23.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -7.103038796341682 \cdot 10^{-23}:\\ \;\;\;\;(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) + \left(-{x}^{\left(\frac{1}{n}\right)}\right))_*\\ \mathbf{elif}\;\frac{1}{n} \le 5.793954106697829 \cdot 10^{-10}:\\ \;\;\;\;(\left(\frac{\frac{1}{4}}{x}\right) \cdot \left(\frac{\log x}{n \cdot n}\right) + \left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{x}\right) \cdot \frac{\frac{1}{x}}{n}\right))_* \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{e^{\frac{\log_* (1 + x)}{n}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{\frac{\log_* (1 + x)}{n}}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{\left({x}^{\left(\frac{1}{n}\right)} \cdot e^{\frac{\log_* (1 + x)}{n}} + {x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right) + e^{\frac{\log_* (1 + x)}{n}} \cdot e^{\frac{\log_* (1 + x)}{n}}}\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018230 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))