Average Error: 33.5 → 24.0
Time: 1.4m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.00011763405955802261:\\ \;\;\;\;\sqrt[3]{\left(\left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left(\left(\sqrt{e^{\frac{\log_* (1 + x)}{n}}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{e^{\frac{\log_* (1 + x)}{n}}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.7902523274654044 \cdot 10^{-09}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right) + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log_* (1 + x)}{n}} \cdot e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}}{e^{\frac{\log_* (1 + x)}{n}} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (/ 1 n) < -0.00011763405955802261

    1. Initial program 0.9

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

    3. Applied add-exp-log0.9

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

    4. Applied pow-exp0.9

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

    5. Simplified0.9

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

    7. Applied add-cbrt-cube0.9

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

    9. Applied add-sqr-sqrt0.9

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

    10. Applied add-sqr-sqrt0.9

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

    11. Applied difference-of-squares0.9

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

    if -0.00011763405955802261 < (/ 1 n) < 1.7902523274654044e-09

    1. Initial program 45.5

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

    3. Applied add-exp-log45.5

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

    4. Applied pow-exp45.5

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

    5. Simplified45.5

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

    7. Applied add-cbrt-cube45.5

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

    8. Taylor expanded around -inf 63.0

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

    9. Simplified32.6

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

    if 1.7902523274654044e-09 < (/ 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-log6.9

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

    4. Applied pow-exp6.9

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

    5. Simplified4.1

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

    7. Applied flip--4.1

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

  4. Final simplification24.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.00011763405955802261:\\ \;\;\;\;\sqrt[3]{\left(\left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left(\left(\sqrt{e^{\frac{\log_* (1 + x)}{n}}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{e^{\frac{\log_* (1 + x)}{n}}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.7902523274654044 \cdot 10^{-09}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right) + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log_* (1 + x)}{n}} \cdot e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}}{e^{\frac{\log_* (1 + x)}{n}} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018277 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))