Average Error: 32.9 → 9.1
Time: 1.7m
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}\;\left(\frac{\log x}{n} \cdot \frac{\frac{1}{4}}{n \cdot x} + \frac{\frac{1}{n}}{x} \cdot \left(\frac{1}{2} - \frac{\frac{1}{4}}{x}\right)\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right) = -\infty:\\ \;\;\;\;\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\left(\frac{\log x}{n} \cdot \frac{\frac{1}{4}}{n \cdot x} + \frac{\frac{1}{n}}{x} \cdot \left(\frac{1}{2} - \frac{\frac{1}{4}}{x}\right)\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right) \le -6.79254651959102 \cdot 10^{-196}:\\ \;\;\;\;\left(\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\log x \cdot \frac{\frac{1}{2}}{n}}{\frac{n}{\log x}}\right) - \frac{\log x}{n}\\ \mathbf{if}\;\left(\frac{\log x}{n} \cdot \frac{\frac{1}{4}}{n \cdot x} + \frac{\frac{1}{n}}{x} \cdot \left(\frac{1}{2} - \frac{\frac{1}{4}}{x}\right)\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right) \le 7.9932758823026 \cdot 10^{-68}:\\ \;\;\;\;\left(\frac{\log x}{n} \cdot \frac{\frac{1}{4}}{n \cdot x} + \frac{\frac{1}{n}}{x} \cdot \left(\frac{1}{2} - \frac{\frac{1}{4}}{x}\right)\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{if}\;\left(\frac{\log x}{n} \cdot \frac{\frac{1}{4}}{n \cdot x} + \frac{\frac{1}{n}}{x} \cdot \left(\frac{1}{2} - \frac{\frac{1}{4}}{x}\right)\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right) \le +\infty:\\ \;\;\;\;\left(\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\log x \cdot \frac{\frac{1}{2}}{n}}{\frac{n}{\log x}}\right) - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 4 regimes

  2. if (* (+ (* (/ (log x) n) (/ 1/4 (* n x))) (* (/ (/ 1 n) x) (- 1/2 (/ 1/4 x)))) (+ (sqrt (pow x (/ 1 n))) (sqrt (pow (+ 1 x) (/ 1 n))))) < -inf.0

    1. Initial program 24.0

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

    3. Applied add-sqr-sqrt24.0

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

    if -inf.0 < (* (+ (* (/ (log x) n) (/ 1/4 (* n x))) (* (/ (/ 1 n) x) (- 1/2 (/ 1/4 x)))) (+ (sqrt (pow x (/ 1 n))) (sqrt (pow (+ 1 x) (/ 1 n))))) < -6.79254651959102e-196 or 7.9932758823026e-68 < (* (+ (* (/ (log x) n) (/ 1/4 (* n x))) (* (/ (/ 1 n) x) (- 1/2 (/ 1/4 x)))) (+ (sqrt (pow x (/ 1 n))) (sqrt (pow (+ 1 x) (/ 1 n))))) < +inf.0

    1. Initial program 56.6

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

    2. Taylor expanded around inf 59.5

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

    3. Applied simplify11.2

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

    if -6.79254651959102e-196 < (* (+ (* (/ (log x) n) (/ 1/4 (* n x))) (* (/ (/ 1 n) x) (- 1/2 (/ 1/4 x)))) (+ (sqrt (pow x (/ 1 n))) (sqrt (pow (+ 1 x) (/ 1 n))))) < 7.9932758823026e-68

    1. Initial program 21.6

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

    3. Applied add-sqr-sqrt21.7

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

    4. Applied add-sqr-sqrt21.6

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

    5. Applied difference-of-squares21.6

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

    6. Taylor expanded around inf 11.1

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

    7. Applied simplify4.9

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

    if +inf.0 < (* (+ (* (/ (log x) n) (/ 1/4 (* n x))) (* (/ (/ 1 n) x) (- 1/2 (/ 1/4 x)))) (+ (sqrt (pow x (/ 1 n))) (sqrt (pow (+ 1 x) (/ 1 n)))))

    1. Initial program 0

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

    3. Applied add-sqr-sqrt0

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))