Average Error: 33.2 → 18.7
Time: 1.7m
Precision: 64
Internal Precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{\frac{1}{x}}{n} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + \frac{\frac{\log x}{x}}{n \cdot n} \le -9.817622073517543 \cdot 10^{-139}:\\ \;\;\;\;\left(\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\log x \cdot \frac{\frac{1}{2}}{n}}{\frac{n}{\log x}}\right) - \frac{\log x}{n}\\ \mathbf{if}\;\left(\frac{\frac{1}{x}}{n} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + \frac{\frac{\log x}{x}}{n \cdot n} \le 4.264924977041202 \cdot 10^{-43}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + \frac{\frac{\log x}{x}}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(e^{\frac{\log \left(x + 1\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (+ (- (/ (/ 1 x) n) (/ (/ 1/2 n) (* x x))) (/ (/ (log x) x) (* n n))) < -9.817622073517543e-139

    1. Initial program 47.7

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

    2. Taylor expanded around inf 60.1

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

    3. Applied simplify18.1

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

    if -9.817622073517543e-139 < (+ (- (/ (/ 1 x) n) (/ (/ 1/2 n) (* x x))) (/ (/ (log x) x) (* n n))) < 4.264924977041202e-43

    1. Initial program 30.6

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

    2. Taylor expanded around -inf 63.0

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

    3. Applied simplify12.8

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

    4. Taylor expanded around -inf 63.0

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

    5. Applied simplify12.8

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

    if 4.264924977041202e-43 < (+ (- (/ (/ 1 x) n) (/ (/ 1/2 n) (* x x))) (/ (/ (log x) x) (* n n)))

    1. Initial program 26.3

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

    3. Applied add-cbrt-cube26.4

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

    4. Applied simplify26.4

      \[\leadsto \sqrt[3]{\color{blue}{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}}\]
    5. Using strategy rm

    6. Applied add-exp-log26.4

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

    7. Applied pow-exp26.4

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

    8. Applied simplify26.4

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))