Average Error: 32.5 → 18.3
Time: 1.7m
Precision: 64
Internal Precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{x \cdot n} - \frac{\frac{\frac{1}{2}}{x}}{x \cdot n}\right) + \frac{\log x}{\left(n \cdot n\right) \cdot x} \le -7.533213007031442 \cdot 10^{-38}:\\ \;\;\;\;\left(\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\log x \cdot \frac{\frac{1}{2}}{n}}{\frac{n}{\log x}}\right) - \frac{\log x}{n}\\ \mathbf{if}\;\left(\frac{1}{x \cdot n} - \frac{\frac{\frac{1}{2}}{x}}{x \cdot n}\right) + \frac{\log x}{\left(n \cdot n\right) \cdot x} \le 2.223130515316725 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{x \cdot n}}{n} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right)\\ \mathbf{if}\;\left(\frac{1}{x \cdot n} - \frac{\frac{\frac{1}{2}}{x}}{x \cdot n}\right) + \frac{\log x}{\left(n \cdot n\right) \cdot x} \le 4.112546575160632 \cdot 10^{+153}:\\ \;\;\;\;\left(\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\log x \cdot \frac{\frac{1}{2}}{n}}{\frac{n}{\log x}}\right) - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (+ (- (/ 1 (* x n)) (/ (/ 1/2 x) (* x n))) (/ (log x) (* (* n n) x))) < -7.533213007031442e-38 or 2.223130515316725e-75 < (+ (- (/ 1 (* x n)) (/ (/ 1/2 x) (* x n))) (/ (log x) (* (* n n) x))) < 4.112546575160632e+153

    1. Initial program 39.7

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

    2. Taylor expanded around inf 60.4

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

    3. Applied simplify22.6

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

    if -7.533213007031442e-38 < (+ (- (/ 1 (* x n)) (/ (/ 1/2 x) (* x n))) (/ (log x) (* (* n n) x))) < 2.223130515316725e-75

    1. Initial program 32.6

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

    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 simplify12.7

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

    5. Applied sub-neg12.7

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

    6. Applied associate-+l+12.7

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

    7. Applied simplify12.7

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

    if 4.112546575160632e+153 < (+ (- (/ 1 (* x n)) (/ (/ 1/2 x) (* x n))) (/ (log x) (* (* n n) x)))

    1. Initial program 22.8

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

    3. Applied add-cbrt-cube22.8

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

    4. Applied simplify22.8

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))