Average Error: 32.7 → 7.1
Time: 2.3m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;(e^{\log \left(1 + x\right) \cdot \frac{1}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le -1.7124961272093298 \cdot 10^{-09}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;(e^{\log \left(1 + x\right) \cdot \frac{1}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le -5.1306154706443805 \cdot 10^{-301}:\\ \;\;\;\;(e^{\log \left(1 + x\right) \cdot \frac{1}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*\\ \mathbf{if}\;(e^{\log \left(1 + x\right) \cdot \frac{1}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le 1.3209667438911694 \cdot 10^{-305}:\\ \;\;\;\;\frac{\frac{\log x}{n \cdot n}}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{1}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;(e^{\log \left(1 + x\right) \cdot \frac{1}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (- (expm1 (* (log (+ 1 x)) (/ 1 n))) (fma (/ 1/2 (* n n)) (* (log x) (log x)) (/ (log x) n))) < -1.7124961272093298e-09

    1. Initial program 1.9

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

    3. Applied add-exp-log1.9

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

    4. Applied pow-exp1.9

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

    5. Applied simplify0.8

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

    if -1.7124961272093298e-09 < (- (expm1 (* (log (+ 1 x)) (/ 1 n))) (fma (/ 1/2 (* n n)) (* (log x) (log x)) (/ (log x) n))) < -5.1306154706443805e-301 or 1.3209667438911694e-305 < (- (expm1 (* (log (+ 1 x)) (/ 1 n))) (fma (/ 1/2 (* n n)) (* (log x) (log x)) (/ (log x) n)))

    1. Initial program 53.9

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

    2. Taylor expanded around inf 54.3

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

    3. Applied simplify15.8

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

    5. Applied add-exp-log15.8

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

    6. Applied pow-exp15.8

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

    7. Applied expm1-def13.7

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

    if -5.1306154706443805e-301 < (- (expm1 (* (log (+ 1 x)) (/ 1 n))) (fma (/ 1/2 (* n n)) (* (log x) (log x)) (/ (log x) n))) < 1.3209667438911694e-305

    1. Initial program 28.2

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

    2. Taylor expanded around inf 2.2

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

    3. Applied simplify2.2

      \[\leadsto \color{blue}{\frac{\frac{\log x}{n \cdot n}}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{1}{x \cdot n}\right)}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed 2018206 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))