2nthrt (problem 3.4.6)

Average Error: 29.3 → 23.0

Time: 30.0s

Precision: 64

• could not determine a ground truth for program body

  1. x = 2.931459133389821e+208
  2. n = 3.625484546781197e-95

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.1987118596397397750669009996408489051 \cdot 10^{-14}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} + \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right) \cdot \left(\left(\sqrt[3]{\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}}\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.610123296819638228104037622201438750835 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{1 \cdot \log x}{x \cdot \left(n \cdot n\right)}\right) - \frac{0.5}{n \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} + \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right) \cdot \frac{\sqrt{\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{\frac{1}{n}}{2}\right)}} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) + \sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}}\right)}\\ \end{array}\]

C

double f(double x, double n) {
        double r4591880 = x;
        double r4591881 = 1.0;
        double r4591882 = r4591880 + r4591881;
        double r4591883 = n;
        double r4591884 = r4591881 / r4591883;
        double r4591885 = pow(r4591882, r4591884);
        double r4591886 = pow(r4591880, r4591884);
        double r4591887 = r4591885 - r4591886;
        return r4591887;
}

↓

double f(double x, double n) {
        double r4591888 = 1.0;
        double r4591889 = n;
        double r4591890 = r4591888 / r4591889;
        double r4591891 = -1.1987118596397398e-14;
        bool r4591892 = r4591890 <= r4591891;
        double r4591893 = x;
        double r4591894 = r4591893 + r4591888;
        double r4591895 = pow(r4591894, r4591890);
        double r4591896 = pow(r4591893, r4591890);
        double r4591897 = r4591895 - r4591896;
        double r4591898 = cbrt(r4591897);
        double r4591899 = r4591898 * r4591898;
        double r4591900 = sqrt(r4591895);
        double r4591901 = 2.0;
        double r4591902 = r4591890 / r4591901;
        double r4591903 = pow(r4591893, r4591902);
        double r4591904 = r4591900 + r4591903;
        double r4591905 = sqrt(r4591900);
        double r4591906 = sqrt(r4591903);
        double r4591907 = r4591905 + r4591906;
        double r4591908 = r4591905 - r4591906;
        double r4591909 = cbrt(r4591908);
        double r4591910 = r4591909 * r4591909;
        double r4591911 = r4591910 * r4591909;
        double r4591912 = r4591907 * r4591911;
        double r4591913 = r4591904 * r4591912;
        double r4591914 = cbrt(r4591913);
        double r4591915 = r4591899 * r4591914;
        double r4591916 = 2.6101232968196382e-06;
        bool r4591917 = r4591890 <= r4591916;
        double r4591918 = r4591893 * r4591889;
        double r4591919 = r4591888 / r4591918;
        double r4591920 = log(r4591893);
        double r4591921 = r4591888 * r4591920;
        double r4591922 = r4591889 * r4591889;
        double r4591923 = r4591893 * r4591922;
        double r4591924 = r4591921 / r4591923;
        double r4591925 = r4591919 + r4591924;
        double r4591926 = 0.5;
        double r4591927 = r4591893 * r4591893;
        double r4591928 = r4591889 * r4591927;
        double r4591929 = r4591926 / r4591928;
        double r4591930 = r4591925 - r4591929;
        double r4591931 = r4591905 * r4591900;
        double r4591932 = r4591906 * r4591903;
        double r4591933 = r4591931 - r4591932;
        double r4591934 = r4591905 * r4591906;
        double r4591935 = r4591904 + r4591934;
        double r4591936 = r4591933 / r4591935;
        double r4591937 = r4591907 * r4591936;
        double r4591938 = r4591904 * r4591937;
        double r4591939 = cbrt(r4591938);
        double r4591940 = r4591899 * r4591939;
        double r4591941 = r4591917 ? r4591930 : r4591940;
        double r4591942 = r4591892 ? r4591915 : r4591941;
        return r4591942;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -1.1987118596397398e-14

   1. Initial program 1.7

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

   3. Applied add-cube-cbrt 1.7

      \[\leadsto \color{blue}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\]
   4. Using strategy rm

   5. Applied sqr-pow 1.7

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

   6. Applied add-sqr-sqrt 1.7

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

   7. Applied difference-of-squares 1.7

      \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}}\]
   8. Using strategy rm

   9. Applied add-sqr-sqrt 1.7

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

   10. Applied add-sqr-sqrt 1.7

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

   11. Applied sqrt-prod 1.7

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

   12. Applied difference-of-squares 1.7

       \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\left(\left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} + \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right) \cdot \left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right)\right)}}\]
   13. Using strategy rm

   14. Applied add-cube-cbrt 1.7

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

   if -1.1987118596397398e-14 < (/ 1.0 n) < 2.6101232968196382e-06

   1. Initial program 44.1

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

   2. Taylor expanded around inf 32.3

      \[\leadsto \color{blue}{1 \cdot \frac{1}{x \cdot n} - \left(1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}} + 0.5 \cdot \frac{1}{{x}^{2} \cdot n}\right)}\]

   3. Simplified 32.3

      \[\leadsto \color{blue}{\left(\frac{1}{x \cdot n} + \frac{1 \cdot \log x}{x \cdot \left(n \cdot n\right)}\right) - \frac{0.5}{n \cdot \left(x \cdot x\right)}}\]

   if 2.6101232968196382e-06 < (/ 1.0 n)

   1. Initial program 24.6

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

   3. Applied add-cube-cbrt 24.6

      \[\leadsto \color{blue}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\]
   4. Using strategy rm

   5. Applied sqr-pow 24.6

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

   6. Applied add-sqr-sqrt 24.6

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

   7. Applied difference-of-squares 24.6

      \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}}\]
   8. Using strategy rm

   9. Applied add-sqr-sqrt 24.6

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

   10. Applied add-sqr-sqrt 24.6

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

   11. Applied sqrt-prod 24.6

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

   12. Applied difference-of-squares 24.6

       \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\left(\left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} + \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right) \cdot \left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right)\right)}}\]
   13. Using strategy rm

   14. Applied flip3-- 28.0

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

   15. Simplified 28.0

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

   16. Simplified 28.0

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

4. Final simplification 23.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.1987118596397397750669009996408489051 \cdot 10^{-14}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} + \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right) \cdot \left(\left(\sqrt[3]{\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}}\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.610123296819638228104037622201438750835 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{1 \cdot \log x}{x \cdot \left(n \cdot n\right)}\right) - \frac{0.5}{n \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} + \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right) \cdot \frac{\sqrt{\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{\frac{1}{n}}{2}\right)}} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) + \sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))