Average Error: 32.3 → 22.3
Time: 1.0m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.1379176804553716 \cdot 10^{-13}:\\ \;\;\;\;(\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}\right) + \left(-{x}^{\left(\frac{1}{n}\right)}\right))_*\\ \mathbf{elif}\;\frac{1}{n} \le 1.0911122087182946 \cdot 10^{-06}:\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{\frac{1}{x}}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{\sqrt{\frac{\log_* (1 + x)}{n}}}\right)}^{\left(\sqrt{\frac{\log_* (1 + x)}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (/ 1 n) < -1.1379176804553716e-13

    1. Initial program 2.8

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

    2. Initial simplification2.8

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

    4. Applied add-cube-cbrt2.9

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

    5. Applied unpow-prod-down2.9

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

    6. Applied fma-neg3.0

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

    if -1.1379176804553716e-13 < (/ 1 n) < 1.0911122087182946e-06

    1. Initial program 44.2

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

    2. Initial simplification44.2

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

    3. Taylor expanded around -inf 63.0

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

    4. Simplified30.5

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

    if 1.0911122087182946e-06 < (/ 1 n)

    1. Initial program 5.3

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

    2. Initial simplification5.3

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

    4. Applied add-exp-log5.3

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

    5. Applied pow-exp5.3

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

    6. Simplified2.1

      \[\leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt2.1

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

    9. Applied exp-prod2.1

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

  4. Final simplification22.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.1379176804553716 \cdot 10^{-13}:\\ \;\;\;\;(\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}\right) + \left(-{x}^{\left(\frac{1}{n}\right)}\right))_*\\ \mathbf{elif}\;\frac{1}{n} \le 1.0911122087182946 \cdot 10^{-06}:\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{\frac{1}{x}}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{\sqrt{\frac{\log_* (1 + x)}{n}}}\right)}^{\left(\sqrt{\frac{\log_* (1 + x)}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018257 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))