Average Error: 32.8 → 7.4
Time: 1.9m
Precision: 64
Internal Precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;(e^{\log \left(1 + x\right) \cdot \frac{1}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le -0.0009160170586057822:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{if}\;(e^{\log \left(1 + x\right) \cdot \frac{1}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le -5.0299008192410476 \cdot 10^{-306}:\\ \;\;\;\;(e^{\left(\log \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right) + \log \left(\sqrt[3]{1 + x}\right)\right) \cdot \frac{1}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*\\ \mathbf{if}\;(e^{\log \left(1 + x\right) \cdot \frac{1}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le 4.829764118837382 \cdot 10^{-301}:\\ \;\;\;\;\frac{\frac{\log x}{n \cdot n}}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{1}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;(e^{\log \left(1 + x\right) \cdot \frac{1}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 4 regimes

  2. if (- (expm1 (* (log (+ 1 x)) (/ 1 n))) (fma (/ 1/2 (* n n)) (* (log x) (log x)) (/ (log x) n))) < -0.0009160170586057822

    1. Initial program 1.6

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

    3. Applied add-log-exp2.0

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

    4. Applied add-log-exp2.0

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

    5. Applied diff-log2.0

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

    6. Applied simplify2.0

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

    if -0.0009160170586057822 < (- (expm1 (* (log (+ 1 x)) (/ 1 n))) (fma (/ 1/2 (* n n)) (* (log x) (log x)) (/ (log x) n))) < -5.0299008192410476e-306

    1. Initial program 53.9

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

    2. Taylor expanded around inf 54.0

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

    3. Applied simplify15.6

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

    5. Applied add-exp-log15.6

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

    6. Applied pow-exp15.6

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

    7. Applied expm1-def13.5

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

    9. Applied add-cube-cbrt13.5

      \[\leadsto (e^{\log \color{blue}{\left(\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right) \cdot \sqrt[3]{1 + x}\right)} \cdot \frac{1}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*\]

    10. Applied log-prod13.0

      \[\leadsto (e^{\color{blue}{\left(\log \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right) + \log \left(\sqrt[3]{1 + x}\right)\right)} \cdot \frac{1}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*\]

    if -5.0299008192410476e-306 < (- (expm1 (* (log (+ 1 x)) (/ 1 n))) (fma (/ 1/2 (* n n)) (* (log x) (log x)) (/ (log x) n))) < 4.829764118837382e-301

    1. Initial program 29.3

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

    2. Taylor expanded around inf 2.3

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

    3. Applied simplify2.3

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

    if 4.829764118837382e-301 < (- (expm1 (* (log (+ 1 x)) (/ 1 n))) (fma (/ 1/2 (* n n)) (* (log x) (log x)) (/ (log x) n)))

    1. Initial program 53.3

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

    2. Taylor expanded around inf 54.2

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

    3. Applied simplify16.0

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

    5. Applied add-exp-log16.0

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

    6. Applied pow-exp16.0

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

    7. Applied expm1-def13.9

      \[\leadsto \color{blue}{(e^{\log \left(1 + x\right) \cdot \frac{1}{n}} - 1)^*} - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 1.9m)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))))