Average Error: 32.8 → 13.5
Time: 2.2m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{n} + \frac{\log x}{n}\right) \cdot \left(\left(\frac{x}{n} - \frac{\log x}{n}\right) \cdot n + \left(\left(x - \log x\right) \cdot \left(\log x + x\right)\right) \cdot \frac{\frac{1}{2}}{n}\right)}{x + \log x} \le -0.09805495681983642:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{if}\;\frac{\left(\frac{x}{n} + \frac{\log x}{n}\right) \cdot \left(\left(\frac{x}{n} - \frac{\log x}{n}\right) \cdot n + \left(\left(x - \log x\right) \cdot \left(\log x + x\right)\right) \cdot \frac{\frac{1}{2}}{n}\right)}{x + \log x} \le 5.951251518515046 \cdot 10^{-16}:\\ \;\;\;\;\left(\frac{x}{n} - \frac{1}{\frac{n}{\log x}}\right) + \frac{\frac{\frac{1}{2}}{n}}{n} \cdot \left(x \cdot x - \log x \cdot \log x\right)\\ \mathbf{if}\;\frac{\left(\frac{x}{n} + \frac{\log x}{n}\right) \cdot \left(\left(\frac{x}{n} - \frac{\log x}{n}\right) \cdot n + \left(\left(x - \log x\right) \cdot \left(\log x + x\right)\right) \cdot \frac{\frac{1}{2}}{n}\right)}{x + \log x} \le 7.733959759287964 \cdot 10^{+184}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + \frac{\log x}{\left(n \cdot x\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 4 regimes

  2. if (/ (* (+ (/ x n) (/ (log x) n)) (+ (* (- (/ x n) (/ (log x) n)) n) (* (* (- x (log x)) (+ (log x) x)) (/ 1/2 n)))) (+ x (log x))) < -0.09805495681983642

    1. Initial program 3.8

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

    3. Applied flip--4.3

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

    if -0.09805495681983642 < (/ (* (+ (/ x n) (/ (log x) n)) (+ (* (- (/ x n) (/ (log x) n)) n) (* (* (- x (log x)) (+ (log x) x)) (/ 1/2 n)))) (+ x (log x))) < 5.951251518515046e-16

    1. Initial program 54.7

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

    2. Taylor expanded around inf 54.9

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

    3. Applied simplify18.3

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

    4. Taylor expanded around 0 17.8

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

    5. Applied simplify17.8

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

    7. Applied clear-num17.8

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

    if 5.951251518515046e-16 < (/ (* (+ (/ x n) (/ (log x) n)) (+ (* (- (/ x n) (/ (log x) n)) n) (* (* (- x (log x)) (+ (log x) x)) (/ 1/2 n)))) (+ x (log x))) < 7.733959759287964e+184

    1. Initial program 47.3

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

    2. Taylor expanded around -inf 62.8

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

    3. Applied simplify19.8

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

    if 7.733959759287964e+184 < (/ (* (+ (/ x n) (/ (log x) n)) (+ (* (- (/ x n) (/ (log x) n)) n) (* (* (- x (log x)) (+ (log x) x)) (/ 1/2 n)))) (+ x (log x)))

    1. Initial program 9.3

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

    3. Applied add-cbrt-cube9.4

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

    4. Applied simplify9.4

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

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed '#(1071119240 1686926585 3481876196 78132896 2080707795 3185793749)' 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))