Average Error: 32.4 → 22.5
Time: 2.0m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(1 + \left(\frac{1}{x \cdot n} + \frac{\log x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \le -4.1076371390619497 \cdot 10^{-224}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - e^{\log \left({x}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{elif}\;\left(1 + \left(\frac{1}{x \cdot n} + \frac{\log x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \le 1.5280632478110775 \cdot 10^{-96}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{1}{2}}{\log \left(e^{n \cdot \left(x \cdot x\right)}\right)}\right) - \frac{-\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left({\left(1 + x\right)}^{\left(\frac{3}{n}\right)} - {x}^{\left(\frac{3}{n}\right)}\right) - \log \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{1}{n}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)\right)}\\ \end{array}\]

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 (+ (/ 1 (* n x)) (/ (log x) n))) (pow x (/ 1 n))) < -4.1076371390619497e-224

    1. Initial program 20.8

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

    3. Applied add-exp-log20.9

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

    if -4.1076371390619497e-224 < (- (+ 1 (+ (/ 1 (* n x)) (/ (log x) n))) (pow x (/ 1 n))) < 1.5280632478110775e-96

    1. Initial program 39.3

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

    2. Taylor expanded around inf 21.5

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

    3. Simplified20.7

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

    5. Applied add-log-exp20.6

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

    if 1.5280632478110775e-96 < (- (+ 1 (+ (/ 1 (* n x)) (/ (log x) n))) (pow x (/ 1 n)))

    1. Initial program 30.7

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

    3. Applied add-exp-log30.7

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

    5. Applied flip3--30.7

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

    6. Applied log-div30.7

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

    7. Simplified30.6

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

  4. Final simplification22.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \left(\frac{1}{x \cdot n} + \frac{\log x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \le -4.1076371390619497 \cdot 10^{-224}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - e^{\log \left({x}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{elif}\;\left(1 + \left(\frac{1}{x \cdot n} + \frac{\log x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \le 1.5280632478110775 \cdot 10^{-96}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{1}{2}}{\log \left(e^{n \cdot \left(x \cdot x\right)}\right)}\right) - \frac{-\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left({\left(1 + x\right)}^{\left(\frac{3}{n}\right)} - {x}^{\left(\frac{3}{n}\right)}\right) - \log \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{1}{n}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018214 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))