Average Error: 33.1 → 23.0
Time: 1.2m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(1 + \frac{\log x}{n}\right) + \frac{1}{x \cdot n}\right) - {x}^{\left(\frac{1}{n}\right)} \le -0.010295450963190445:\\ \;\;\;\;(\left({\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right) + \left(-{x}^{\left(\frac{1}{n}\right)}\right))_*\\ \mathbf{elif}\;\left(\left(1 + \frac{\log x}{n}\right) + \frac{1}{x \cdot n}\right) - {x}^{\left(\frac{1}{n}\right)} \le 5.206381854628619 \cdot 10^{-296}:\\ \;\;\;\;(\left(\frac{1}{x \cdot n}\right) \cdot \left(-\frac{\frac{1}{2}}{x}\right) + \left(\frac{1}{x \cdot n}\right))_* + \frac{\frac{\log x}{x \cdot n}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (- (+ (/ 1 (* n x)) (+ 1 (/ (log x) n))) (pow x (/ 1 n))) < -0.010295450963190445

    1. Initial program 19.8

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

    3. Applied add-cube-cbrt19.9

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

    4. Applied add-cube-cbrt19.9

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

    5. Applied unpow-prod-down19.9

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

    6. Applied prod-diff19.9

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

    7. Simplified19.8

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

    8. Simplified19.8

      \[\leadsto (\left({\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right) + \left(-{x}^{\left(\frac{1}{n}\right)}\right))_* + \color{blue}{0}\]

    if -0.010295450963190445 < (- (+ (/ 1 (* n x)) (+ 1 (/ (log x) n))) (pow x (/ 1 n))) < 5.206381854628619e-296

    1. Initial program 40.7

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

    2. Taylor expanded around inf 22.3

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

    3. Simplified22.3

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

    if 5.206381854628619e-296 < (- (+ (/ 1 (* n x)) (+ 1 (/ (log x) n))) (pow x (/ 1 n)))

    1. Initial program 31.2

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

    3. Applied add-exp-log31.2

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

    4. Applied pow-exp31.2

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

    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 simplification23.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(1 + \frac{\log x}{n}\right) + \frac{1}{x \cdot n}\right) - {x}^{\left(\frac{1}{n}\right)} \le -0.010295450963190445:\\ \;\;\;\;(\left({\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right) + \left(-{x}^{\left(\frac{1}{n}\right)}\right))_*\\ \mathbf{elif}\;\left(\left(1 + \frac{\log x}{n}\right) + \frac{1}{x \cdot n}\right) - {x}^{\left(\frac{1}{n}\right)} \le 5.206381854628619 \cdot 10^{-296}:\\ \;\;\;\;(\left(\frac{1}{x \cdot n}\right) \cdot \left(-\frac{\frac{1}{2}}{x}\right) + \left(\frac{1}{x \cdot n}\right))_* + \frac{\frac{\log x}{x \cdot n}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018217 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))