Average Error: 29.4 → 19.8
Time: 44.2s
Precision: 64
Internal Precision: 128
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -206.4372690337579:\\ \;\;\;\;\log \left(e^{\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}}\right) + \left(\frac{\frac{1}{x}}{n} + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\\ \mathbf{elif}\;n \le -3.4623818328224 \cdot 10^{-310}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - e^{\log \left({x}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{elif}\;n \le 5.23549665628974 \cdot 10^{+44}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - e^{\log \left({x}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} + \frac{\sqrt{\log x}}{\frac{n \cdot \left(x \cdot n\right)}{\sqrt{\log x}}}\right) + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\\ \end{array}\]

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 4 regimes

  2. if n < -206.4372690337579

    1. Initial program 44.3

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

    2. Initial simplification44.3

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

    4. Applied add-exp-log44.3

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

    5. Taylor expanded around -inf 63.2

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

    6. Simplified32.2

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

    8. Applied add-log-exp32.2

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

    if -206.4372690337579 < n < -3.4623818328224e-310

    1. Initial program 0.2

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

    2. Initial simplification0.2

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

    4. Applied add-exp-log0.2

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

    if -3.4623818328224e-310 < n < 5.23549665628974e+44

    1. Initial program 31.3

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

    2. Initial simplification31.3

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

    4. Applied add-exp-log31.3

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

    6. Applied add-exp-log31.3

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

    7. Applied pow-exp31.3

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

    8. Simplified13.5

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

    if 5.23549665628974e+44 < n

    1. Initial program 43.4

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

    2. Initial simplification43.4

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

    4. Applied add-exp-log43.4

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

    5. Taylor expanded around -inf 62.8

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

    6. Simplified31.4

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

    8. Applied add-sqr-sqrt31.4

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

    9. Applied associate-/l*31.4

      \[\leadsto \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\left(\frac{\frac{1}{x}}{n} + 0\right) + \color{blue}{\frac{\sqrt{\log x}}{\frac{n \cdot \left(x \cdot n\right)}{\sqrt{\log x}}}}\right)\]
  3. Recombined 4 regimes into one program.

  4. Final simplification19.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -206.4372690337579:\\ \;\;\;\;\log \left(e^{\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}}\right) + \left(\frac{\frac{1}{x}}{n} + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\\ \mathbf{elif}\;n \le -3.4623818328224 \cdot 10^{-310}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - e^{\log \left({x}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{elif}\;n \le 5.23549665628974 \cdot 10^{+44}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - e^{\log \left({x}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} + \frac{\sqrt{\log x}}{\frac{n \cdot \left(x \cdot n\right)}{\sqrt{\log x}}}\right) + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\\ \end{array}\]

Runtime

Time bar (total: 44.2s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.419.817.511.981.2%
herbie shell --seed 2018351 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))