Average Error: 32.8 → 2.1
Time: 1.6m
Precision: 64
Internal Precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le -1.8606503413718102 \cdot 10^{-08}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le -1.2624064801232432 \cdot 10^{-305}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*\\ \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le 0.0:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\right) + \frac{\frac{\log x}{x}}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (- (expm1 (/ (log1p x) n)) (fma (/ 1/2 (* n n)) (* (log x) (log x)) (/ (log x) n))) < -1.8606503413718102e-08

    1. Initial program 1.9

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

    3. Applied add-exp-log1.9

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

    4. Applied pow-exp1.9

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

    5. Applied simplify0.9

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

    if -1.8606503413718102e-08 < (- (expm1 (/ (log1p x) n)) (fma (/ 1/2 (* n n)) (* (log x) (log x)) (/ (log x) n))) < -1.2624064801232432e-305 or 0.0 < (- (expm1 (/ (log1p x) n)) (fma (/ 1/2 (* n n)) (* (log x) (log x)) (/ (log x) n)))

    1. Initial program 58.8

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

    3. Applied add-exp-log58.8

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

    4. Applied pow-exp58.8

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

    5. Applied simplify58.8

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

    6. Taylor expanded around inf 59.4

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

    7. Applied simplify3.4

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

    if -1.2624064801232432e-305 < (- (expm1 (/ (log1p x) n)) (fma (/ 1/2 (* n n)) (* (log x) (log x)) (/ (log x) n))) < 0.0

    1. Initial program 29.4

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

    2. Taylor expanded around inf 1.5

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

    3. Applied simplify1.5

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))