Average Error: 32.9 → 24.0
Time: 1.7m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.595877941235969 \cdot 10^{-06}:\\ \;\;\;\;\sqrt[3]{\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.8042377868253832 \cdot 10^{-24}:\\ \;\;\;\;\left(\frac{-\frac{\frac{1}{2}}{x}}{x \cdot n} + \frac{\frac{\log x}{x \cdot n}}{n}\right) + \frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}}\right)\right) \cdot {\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (/ 1 n) < -3.595877941235969e-06

    1. Initial program 1.0

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

    2. Initial simplification1.0

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

    4. Applied add-cbrt-cube1.2

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

    6. Applied add-sqr-sqrt1.2

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

    7. Applied add-sqr-sqrt1.2

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

    8. Applied difference-of-squares1.2

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

    if -3.595877941235969e-06 < (/ 1 n) < 1.8042377868253832e-24

    1. Initial program 45.0

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

    2. Initial simplification45.0

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

    4. Applied add-cube-cbrt45.0

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

    5. Applied unpow-prod-down45.0

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

    6. Taylor expanded around -inf 63.0

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

    7. Simplified32.1

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

    if 1.8042377868253832e-24 < (/ 1 n)

    1. Initial program 14.5

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

    2. Initial simplification14.5

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

    4. Applied add-cube-cbrt14.5

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

    5. Applied unpow-prod-down14.5

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

    7. Applied add-cube-cbrt14.6

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

  4. Final simplification24.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.595877941235969 \cdot 10^{-06}:\\ \;\;\;\;\sqrt[3]{\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.8042377868253832 \cdot 10^{-24}:\\ \;\;\;\;\left(\frac{-\frac{\frac{1}{2}}{x}}{x \cdot n} + \frac{\frac{\log x}{x \cdot n}}{n}\right) + \frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}}\right)\right) \cdot {\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018234 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))