Average Error: 32.8 → 2.9
Time: 2.8m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le -9.3992157509438 \cdot 10^{-09}:\\ \;\;\;\;\log \left(e^{\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\right)\\ \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le -3.2291770987512184 \cdot 10^{-292}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*\\ \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le 0.0:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\right) + \frac{\frac{\log x}{x}}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (- (expm1 (/ (log1p x) n)) (fma (/ 1/2 (* n n)) (* (log x) (log x)) (/ (log x) n))) < -9.3992157509438e-09

    1. Initial program 2.1

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

    3. Applied add-log-exp2.4

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

    4. Applied add-log-exp2.3

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

    5. Applied diff-log2.3

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

    6. Applied simplify2.3

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

    8. Applied add-sqr-sqrt2.4

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

    9. Applied add-sqr-sqrt2.4

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

    10. Applied unpow-prod-down2.4

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

    11. Applied difference-of-squares2.3

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

    if -9.3992157509438e-09 < (- (expm1 (/ (log1p x) n)) (fma (/ 1/2 (* n n)) (* (log x) (log x)) (/ (log x) n))) < -3.2291770987512184e-292 or 0.0 < (- (expm1 (/ (log1p x) n)) (fma (/ 1/2 (* n n)) (* (log x) (log x)) (/ (log x) n)))

    1. Initial program 59.0

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

    3. Applied add-exp-log59.0

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

    4. Applied pow-exp59.0

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

    5. Applied simplify59.0

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

    6. Taylor expanded around inf 59.5

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

    7. Applied simplify3.3

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

    if -3.2291770987512184e-292 < (- (expm1 (/ (log1p x) n)) (fma (/ 1/2 (* n n)) (* (log x) (log x)) (/ (log x) n))) < 0.0

    1. Initial program 29.4

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

    2. Taylor expanded around inf 3.1

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

    3. Applied simplify3.1

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

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))