Average Error: 32.3 → 17.0
Time: 1.1m
Precision: 64
Internal Precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 4.512235498273315 \cdot 10^{-293}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}\\ \mathbf{if}\;x \le 0.6644846584934392:\\ \;\;\;\;\left(\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\log x \cdot \frac{\frac{1}{2}}{n}}{\frac{n}{\log x}}\right) - \frac{\log x}{n}\\ \mathbf{if}\;x \le 1.0233214133709562 \cdot 10^{+85}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + \frac{\log x}{\left(n \cdot x\right) \cdot n}\\ \mathbf{if}\;x \le 7.397169367886428 \cdot 10^{+148}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}\\ \mathbf{if}\;x \le 1.0602671418134844 \cdot 10^{+160}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + \frac{\log x}{\left(n \cdot x\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if x < 4.512235498273315e-293 or 1.0233214133709562e+85 < x < 7.397169367886428e+148 or 1.0602671418134844e+160 < x

    1. Initial program 16.9

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

    3. Applied pow-to-exp16.9

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

    4. Applied simplify16.9

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

    if 4.512235498273315e-293 < x < 0.6644846584934392

    1. Initial program 47.2

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

    2. Taylor expanded around inf 60.0

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

    3. Applied simplify14.7

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

    if 0.6644846584934392 < x < 1.0233214133709562e+85 or 7.397169367886428e+148 < x < 1.0602671418134844e+160

    1. Initial program 33.1

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

    2. Taylor expanded around -inf 63.0

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

    3. Applied simplify22.6

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))