Average Error: 32.4 → 22.8
Time: 4.9m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\frac{\log x}{n} + 1\right) + \frac{1}{n \cdot x}\right) - {x}^{\left(\frac{1}{n}\right)} \le -7.782920819837074:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;\left(\left(\frac{\log x}{n} + 1\right) + \frac{1}{n \cdot x}\right) - {x}^{\left(\frac{1}{n}\right)} \le 2.198958655244897 \cdot 10^{-308}:\\ \;\;\;\;\frac{\frac{\log x}{n \cdot n}}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{1}{n \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (- (+ (/ 1 (* x n)) (+ (/ (log x) n) 1)) (pow x (/ 1 n))) < -7.782920819837074

    1. Initial program 18.6

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

    3. Applied add-sqr-sqrt18.6

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

    if -7.782920819837074 < (- (+ (/ 1 (* x n)) (+ (/ (log x) n) 1)) (pow x (/ 1 n))) < 2.198958655244897e-308

    1. Initial program 39.9

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

    2. Taylor expanded around inf 22.4

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

    3. Simplified22.3

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

    if 2.198958655244897e-308 < (- (+ (/ 1 (* x n)) (+ (/ (log x) n) 1)) (pow x (/ 1 n)))

    1. Initial program 31.3

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

    3. Applied add-exp-log31.3

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

    4. Applied pow-exp31.3

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

    5. Simplified29.7

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

  4. Final simplification22.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{\log x}{n} + 1\right) + \frac{1}{n \cdot x}\right) - {x}^{\left(\frac{1}{n}\right)} \le -7.782920819837074:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;\left(\left(\frac{\log x}{n} + 1\right) + \frac{1}{n \cdot x}\right) - {x}^{\left(\frac{1}{n}\right)} \le 2.198958655244897 \cdot 10^{-308}:\\ \;\;\;\;\frac{\frac{\log x}{n \cdot n}}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{1}{n \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Runtime

Time bar (total: 4.9m)Debug logProfile

herbie shell --seed 2018215 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))