Average Error: 32.5 → 3.5
Time: 1.5m
Precision: 64
Internal Precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_* \le -5.321328334255843 \cdot 10^{-26}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_* \le -2.0502360328626817 \cdot 10^{-286}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_*\\ \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_* \le 0.0:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \frac{-\log x}{\left(n \cdot n\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (- (expm1 (/ (log1p x) n)) (fma (/ (log x) n) (/ (log x) (/ n 1/2)) (/ (log x) n))) < -5.321328334255843e-26

    1. Initial program 4.5

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

    3. Applied add-exp-log4.6

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

    4. Applied pow-exp4.6

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

    5. Applied simplify3.2

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

    if -5.321328334255843e-26 < (- (expm1 (/ (log1p x) n)) (fma (/ (log x) n) (/ (log x) (/ n 1/2)) (/ (log x) n))) < -2.0502360328626817e-286 or 0.0 < (- (expm1 (/ (log1p x) n)) (fma (/ (log x) n) (/ (log x) (/ n 1/2)) (/ (log x) n)))

    1. Initial program 59.7

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

    3. Applied add-exp-log59.7

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

    4. Applied pow-exp59.7

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

    5. Applied simplify59.6

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

    6. Taylor expanded around inf 59.9

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

    7. Applied simplify2.7

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

    if -2.0502360328626817e-286 < (- (expm1 (/ (log1p x) n)) (fma (/ (log x) n) (/ (log x) (/ n 1/2)) (/ (log x) n))) < 0.0

    1. Initial program 28.5

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

    2. Taylor expanded around inf 4.5

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

    3. Applied simplify4.5

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

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))