Average Error: 32.8 → 13.3
Time: 2.1m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{x}{n} - \frac{\log x}{n}\right) + \frac{\frac{\frac{1}{2}}{n}}{n} \cdot \left(x \cdot x - \log x \cdot \log x\right) \le -5.350026176355435 \cdot 10^{-07}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{if}\;\left(\frac{x}{n} - \frac{\log x}{n}\right) + \frac{\frac{\frac{1}{2}}{n}}{n} \cdot \left(x \cdot x - \log x \cdot \log x\right) \le 0.03719377217694466:\\ \;\;\;\;\left(\frac{x}{n} - \frac{\log x}{n}\right) + \frac{\frac{\frac{1}{2}}{n}}{n} \cdot \left(x \cdot x - \log x \cdot \log x\right)\\ \mathbf{if}\;\left(\frac{x}{n} - \frac{\log x}{n}\right) + \frac{\frac{\frac{1}{2}}{n}}{n} \cdot \left(x \cdot x - \log x \cdot \log x\right) \le 9.872914403887622 \cdot 10^{+123}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{\frac{\frac{1}{2}}{x}}{x \cdot n}\right) + \frac{\log x}{\left(n \cdot n\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 4 regimes

  2. if (+ (- (/ x n) (/ (log x) n)) (* (/ (/ 1/2 n) n) (- (* x x) (* (log x) (log x))))) < -5.350026176355435e-07

    1. Initial program 4.6

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

    3. Applied flip--5.0

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

    if -5.350026176355435e-07 < (+ (- (/ x n) (/ (log x) n)) (* (/ (/ 1/2 n) n) (- (* x x) (* (log x) (log x))))) < 0.03719377217694466

    1. Initial program 55.0

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

    2. Taylor expanded around inf 55.1

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

    3. Applied simplify17.9

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

    4. Taylor expanded around 0 17.4

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

    5. Applied simplify17.4

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

    if 0.03719377217694466 < (+ (- (/ x n) (/ (log x) n)) (* (/ (/ 1/2 n) n) (- (* x x) (* (log x) (log x))))) < 9.872914403887622e+123

    1. Initial program 51.4

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

    2. Taylor expanded around inf 11.2

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

    3. Applied simplify11.2

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

    if 9.872914403887622e+123 < (+ (- (/ x n) (/ (log x) n)) (* (/ (/ 1/2 n) n) (- (* x x) (* (log x) (log x)))))

    1. Initial program 10.4

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

    3. Applied add-sqr-sqrt10.5

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

    4. Applied add-sqr-sqrt10.5

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

    5. Applied unpow-prod-down10.4

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

    6. Applied difference-of-squares10.4

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

    8. Applied add-sqr-sqrt10.4

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed '#(1072107073 2127697367 3936270018 2300570620 2134894798 4023771849)' 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))