Average Error: 32.4 → 2.1
Time: 1.4m
Precision: 64
Internal Precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_*} \cdot \sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_*}\right) \cdot \sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_*} \le -0.7641534621151085:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\left(\sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_*} \cdot \sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_*}\right) \cdot \sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_*} \le -8.295411842152295 \cdot 10^{-306}:\\ \;\;\;\;(\left(\sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^*} \cdot \sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^*}\right) \cdot \left(\sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^*}\right) + \left(-(\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_*\right))_*\\ \mathbf{if}\;\left(\sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_*} \cdot \sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_*}\right) \cdot \sqrt[3]{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_*} \le 0.0:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \frac{-\log x}{\left(n \cdot n\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{1}{\frac{n}{\log x}}\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 4 regimes

  2. if (* (* (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (log x) n) (/ (log x) (/ n 1/2)) (/ (log x) n)))) (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (log x) n) (/ (log x) (/ n 1/2)) (/ (log x) n))))) (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (log x) n) (/ (log x) (/ n 1/2)) (/ (log x) n))))) < -0.7641534621151085

    1. Initial program 1.4

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

    3. Applied add-exp-log1.4

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

    4. Applied pow-exp1.4

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

    5. Applied simplify0.4

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

    if -0.7641534621151085 < (* (* (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (log x) n) (/ (log x) (/ n 1/2)) (/ (log x) n)))) (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (log x) n) (/ (log x) (/ n 1/2)) (/ (log x) n))))) (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (log x) n) (/ (log x) (/ n 1/2)) (/ (log x) n))))) < -8.295411842152295e-306

    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 59.4

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

    7. Applied simplify2.9

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

    9. Applied add-cube-cbrt2.9

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

    10. Applied fma-neg3.0

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

    if -8.295411842152295e-306 < (* (* (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (log x) n) (/ (log x) (/ n 1/2)) (/ (log x) n)))) (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (log x) n) (/ (log x) (/ n 1/2)) (/ (log x) n))))) (cbrt (- (expm1 (/ (log1p x) n)) (fma (/ (log x) n) (/ (log x) (/ n 1/2)) (/ (log x) n))))) < 0.0

    1. Initial program 28.0

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

    2. Taylor expanded around inf 1.6

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

    3. Applied simplify1.6

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

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

    1. Initial program 58.5

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

    3. Applied add-exp-log58.5

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

    4. Applied pow-exp58.5

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

    5. Applied simplify58.4

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

    6. Taylor expanded around inf 59.3

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

    7. Applied simplify4.3

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

    9. Applied clear-num4.5

      \[\leadsto (e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \color{blue}{\left(\frac{1}{\frac{n}{\log x}}\right)})_*\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))