Average Error: 30.5 → 6.7
Time: 1.2m
Precision: 64
Ground Truth: 128
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
⬇
\[\begin{array}{l} \mathbf{if}\;n \le -1626677269.8521662:\\ \;\;\;\;\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\frac{\frac{\frac{1}{2}}{x}}{n} - \left(\frac{\log x \cdot \frac{\frac{1}{4}}{x}}{n \cdot n} + \frac{\frac{\frac{1}{4}}{n}}{{x}^2}\right)\right)\\ \mathbf{if}\;n \le 64407249603897.836:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\frac{\frac{\frac{1}{2}}{x}}{n} - \left(\frac{\log x \cdot \frac{\frac{1}{4}}{x}}{n \cdot n} + \frac{\frac{\frac{1}{4}}{n}}{{x}^2}\right)\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes.

  2. if n < -1626677269.8521662 or 64407249603897.836 < n

    1. Initial program 43.4

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

    3. Applied add-sqr-sqrt 43.4

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

    4. Applied add-sqr-sqrt 43.4

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

    5. Applied difference-of-squares 43.4

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

    6. Applied taylor 8.7

      \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{x \cdot n} - \left(\frac{1}{4} \cdot \frac{\log x}{x \cdot {n}^2} + \frac{1}{4} \cdot \frac{1}{{x}^2 \cdot n}\right)\right)\]

    7. Taylor expanded around inf 8.7

      \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{x \cdot n} - \left(\frac{1}{4} \cdot \frac{\log x}{x \cdot {n}^2} + \frac{1}{4} \cdot \frac{1}{{x}^2 \cdot n}\right)\right)}\]

    8. Applied simplify 8.0

      \[\leadsto \color{blue}{\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\frac{\frac{\frac{1}{2}}{x}}{n} - \left(\frac{\frac{1}{4}}{n} \cdot \frac{\log x}{x \cdot n} + \frac{\frac{\frac{1}{4}}{n}}{x \cdot x}\right)\right)}\]

    9. Applied simplify 8.0

      \[\leadsto \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \color{blue}{\left(\frac{\frac{\frac{1}{2}}{x}}{n} - \left(\frac{\log x \cdot \frac{\frac{1}{4}}{x}}{n \cdot n} + \frac{\frac{\frac{1}{4}}{n}}{{x}^2}\right)\right)}\]

    if -1626677269.8521662 < n < 64407249603897.836

    1. Initial program 4.2

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

    3. Applied add-sqr-sqrt 4.2

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

    4. Applied add-sqr-sqrt 4.2

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

    5. Applied difference-of-squares 4.2

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

    7. Applied add-log-exp 4.3

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

    8. Applied add-log-exp 4.3

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

    9. Applied diff-log 4.3

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

    10. Applied simplify 4.3

      \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \color{blue}{\left(e^{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Total time: 1.2m Debug log

Please report a bug with the following info:

herbie --seed '#(1103931650 2496750099 1453395308 2761556557 1822167000 2886626694)'
(FPCore (x n)
  :name "NMSE problem 3.4.6"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))