Average Error: 33.0 → 22.6
Time: 4.1m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\log \left(e^{\frac{\log x}{n \cdot n}}\right)}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{1}{x \cdot n}\right) \le -1.9350288057299238 \cdot 10^{-20}:\\ \;\;\;\;\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \frac{{\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}{(\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) + \left((\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right) + \left({x}^{\left(\frac{1}{n}\right)}\right))_*\right))_*}\\ \mathbf{if}\;\frac{\log \left(e^{\frac{\log x}{n \cdot n}}\right)}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{1}{x \cdot n}\right) \le 7.797286451813034 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{\log x}{n \cdot n}}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{\frac{1}{n}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (- (/ (log (exp (/ (log x) (* n n)))) x) (- (/ (/ 1/2 n) (* x x)) (/ 1 (* x n)))) < -1.9350288057299238e-20

    1. Initial program 45.2

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

    3. Applied add-sqr-sqrt45.3

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

    4. Applied add-sqr-sqrt45.3

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

    5. Applied unpow-prod-down45.3

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

    6. Applied difference-of-squares45.3

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

    8. Applied flip3--45.3

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

    9. Applied simplify45.3

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

    if -1.9350288057299238e-20 < (- (/ (log (exp (/ (log x) (* n n)))) x) (- (/ (/ 1/2 n) (* x x)) (/ 1 (* x n)))) < 7.797286451813034e-28

    1. Initial program 35.8

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

    2. Taylor expanded around inf 13.1

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

    3. Applied simplify13.1

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

    5. Applied *-un-lft-identity13.1

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

    6. Applied add-sqr-sqrt19.9

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

    7. Applied prod-diff19.9

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

    8. Applied simplify12.2

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

    9. Applied simplify12.2

      \[\leadsto \frac{\frac{\log x}{n \cdot n}}{x} - \left(\left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{\frac{1}{n}}{x}\right) + \color{blue}{0}\right)\]

    if 7.797286451813034e-28 < (- (/ (log (exp (/ (log x) (* n n)))) x) (- (/ (/ 1/2 n) (* x x)) (/ 1 (* x n))))

    1. Initial program 23.2

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

    3. Applied add-log-exp23.4

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

    4. Applied add-log-exp23.4

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

    5. Applied diff-log23.4

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

    6. Applied simplify23.4

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

  4. Applied simplify22.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\log \left(e^{\frac{\log x}{n \cdot n}}\right)}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{1}{x \cdot n}\right) \le -1.9350288057299238 \cdot 10^{-20}:\\ \;\;\;\;\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \frac{{\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}{(\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) + \left((\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right) + \left({x}^{\left(\frac{1}{n}\right)}\right))_*\right))_*}\\ \mathbf{if}\;\frac{\log \left(e^{\frac{\log x}{n \cdot n}}\right)}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{1}{x \cdot n}\right) \le 7.797286451813034 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{\log x}{n \cdot n}}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{\frac{1}{n}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}}\]

Runtime

Time bar (total: 4.1m)Debug logProfile

herbie shell --seed 2018178 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))