Average Error: 32.2 → 21.8
Time: 1.7m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{1}{n}}{x} + \left(\frac{\log x}{n} + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right)\right) \le -0.0011321014337581812:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{if}\;\frac{\frac{1}{n}}{x} + \left(\frac{\log x}{n} + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right)\right) \le 2.445822097022024 \cdot 10^{-16}:\\ \;\;\;\;\left(\frac{\frac{\log x}{x}}{n \cdot n} + \left(\frac{\frac{1}{x}}{n} - \frac{\frac{\frac{1}{2}}{x}}{x \cdot n}\right)\right) + (\left(\frac{-\sqrt[3]{\frac{\frac{1}{2}}{x}}}{x}\right) \cdot \left(\frac{\sqrt[3]{\frac{\frac{1}{2}}{x}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{x}}}{n}\right) + \left(\frac{\sqrt[3]{\frac{\frac{1}{2}}{x}}}{x} \cdot \frac{\sqrt[3]{\frac{\frac{1}{2}}{x}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{x}}}{n}\right))_*\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (+ (/ (/ 1 n) x) (+ (+ 0 (/ (log x) n)) (- 1 (pow x (/ 1 n))))) < -0.0011321014337581812

    1. Initial program 19.3

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

    3. Applied add-log-exp19.6

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

    4. Applied add-log-exp19.5

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

    5. Applied diff-log19.5

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

    6. Applied simplify19.5

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

    if -0.0011321014337581812 < (+ (/ (/ 1 n) x) (+ (+ 0 (/ (log x) n)) (- 1 (pow x (/ 1 n))))) < 2.445822097022024e-16

    1. Initial program 40.2

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

    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 simplify21.5

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

    5. Applied add-cube-cbrt21.5

      \[\leadsto \left(\frac{1}{n \cdot x} + 0\right) + \left(\frac{\frac{\log x}{x}}{n \cdot n} - \frac{\color{blue}{\left(\sqrt[3]{\frac{\frac{1}{2}}{x}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{x}}\right) \cdot \sqrt[3]{\frac{\frac{1}{2}}{x}}}}{n \cdot x}\right)\]

    6. Applied times-frac21.5

      \[\leadsto \left(\frac{1}{n \cdot x} + 0\right) + \left(\frac{\frac{\log x}{x}}{n \cdot n} - \color{blue}{\frac{\sqrt[3]{\frac{\frac{1}{2}}{x}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{x}}}{n} \cdot \frac{\sqrt[3]{\frac{\frac{1}{2}}{x}}}{x}}\right)\]

    7. Applied add-sqr-sqrt21.6

      \[\leadsto \left(\frac{1}{n \cdot x} + 0\right) + \left(\color{blue}{\sqrt{\frac{\frac{\log x}{x}}{n \cdot n}} \cdot \sqrt{\frac{\frac{\log x}{x}}{n \cdot n}}} - \frac{\sqrt[3]{\frac{\frac{1}{2}}{x}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{x}}}{n} \cdot \frac{\sqrt[3]{\frac{\frac{1}{2}}{x}}}{x}\right)\]

    8. Applied prod-diff21.6

      \[\leadsto \left(\frac{1}{n \cdot x} + 0\right) + \color{blue}{\left((\left(\sqrt{\frac{\frac{\log x}{x}}{n \cdot n}}\right) \cdot \left(\sqrt{\frac{\frac{\log x}{x}}{n \cdot n}}\right) + \left(-\frac{\sqrt[3]{\frac{\frac{1}{2}}{x}}}{x} \cdot \frac{\sqrt[3]{\frac{\frac{1}{2}}{x}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{x}}}{n}\right))_* + (\left(-\frac{\sqrt[3]{\frac{\frac{1}{2}}{x}}}{x}\right) \cdot \left(\frac{\sqrt[3]{\frac{\frac{1}{2}}{x}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{x}}}{n}\right) + \left(\frac{\sqrt[3]{\frac{\frac{1}{2}}{x}}}{x} \cdot \frac{\sqrt[3]{\frac{\frac{1}{2}}{x}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{x}}}{n}\right))_*\right)}\]

    9. Applied associate-+r+21.6

      \[\leadsto \color{blue}{\left(\left(\frac{1}{n \cdot x} + 0\right) + (\left(\sqrt{\frac{\frac{\log x}{x}}{n \cdot n}}\right) \cdot \left(\sqrt{\frac{\frac{\log x}{x}}{n \cdot n}}\right) + \left(-\frac{\sqrt[3]{\frac{\frac{1}{2}}{x}}}{x} \cdot \frac{\sqrt[3]{\frac{\frac{1}{2}}{x}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{x}}}{n}\right))_*\right) + (\left(-\frac{\sqrt[3]{\frac{\frac{1}{2}}{x}}}{x}\right) \cdot \left(\frac{\sqrt[3]{\frac{\frac{1}{2}}{x}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{x}}}{n}\right) + \left(\frac{\sqrt[3]{\frac{\frac{1}{2}}{x}}}{x} \cdot \frac{\sqrt[3]{\frac{\frac{1}{2}}{x}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{x}}}{n}\right))_*}\]

    10. Applied simplify20.8

      \[\leadsto \color{blue}{\left(\frac{\frac{\log x}{x}}{n \cdot n} + \left(\frac{\frac{1}{x}}{n} - \frac{\frac{\frac{1}{2}}{x}}{x \cdot n}\right)\right)} + (\left(-\frac{\sqrt[3]{\frac{\frac{1}{2}}{x}}}{x}\right) \cdot \left(\frac{\sqrt[3]{\frac{\frac{1}{2}}{x}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{x}}}{n}\right) + \left(\frac{\sqrt[3]{\frac{\frac{1}{2}}{x}}}{x} \cdot \frac{\sqrt[3]{\frac{\frac{1}{2}}{x}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{x}}}{n}\right))_*\]

    if 2.445822097022024e-16 < (+ (/ (/ 1 n) x) (+ (+ 0 (/ (log x) n)) (- 1 (pow x (/ 1 n)))))

    1. Initial program 29.3

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

    3. Applied add-exp-log29.3

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

    4. Applied pow-exp29.3

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

    5. Applied simplify27.7

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

  4. Applied simplify21.8

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{1}{n}}{x} + \left(\frac{\log x}{n} + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right)\right) \le -0.0011321014337581812:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{if}\;\frac{\frac{1}{n}}{x} + \left(\frac{\log x}{n} + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right)\right) \le 2.445822097022024 \cdot 10^{-16}:\\ \;\;\;\;\left(\frac{\frac{\log x}{x}}{n \cdot n} + \left(\frac{\frac{1}{x}}{n} - \frac{\frac{\frac{1}{2}}{x}}{x \cdot n}\right)\right) + (\left(\frac{-\sqrt[3]{\frac{\frac{1}{2}}{x}}}{x}\right) \cdot \left(\frac{\sqrt[3]{\frac{\frac{1}{2}}{x}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{x}}}{n}\right) + \left(\frac{\sqrt[3]{\frac{\frac{1}{2}}{x}}}{x} \cdot \frac{\sqrt[3]{\frac{\frac{1}{2}}{x}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{x}}}{n}\right))_*\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))