Average Error: 31.2 → 8.3
Time: 59.0s
Precision: 64
Internal precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
⬇
\[\begin{array}{l} \mathbf{if}\;n \le -2744589958.7011433:\\ \;\;\;\;\frac{1}{n \cdot x} - \left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} + \frac{\log x}{{n}^2 \cdot x}\right)\\ \mathbf{if}\;n \le 2.6712054680957466 \cdot 10^{+25}:\\ \;\;\;\;e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x} - \left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} + \frac{\log x}{{n}^2 \cdot x}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes.

  2. if n < -2744589958.7011433 or 2.6712054680957466e+25 < n

    1. Initial program 43.4

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

    2. Applied taylor 9.5

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

    3. Taylor expanded around inf 9.5

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

    if -2744589958.7011433 < n < 2.6712054680957466e+25

    1. Initial program 5.9

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

    3. Applied add-exp-log 5.9

      \[\leadsto \color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 59.0s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(2277612311 2645429965 1090895633 2857793080 2144184008 3989768357)'
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))