2nthrt (problem 3.4.6)

Average Error: 29.4 → 22.6

Time: 32.9s

Precision: 64

• could not determine a ground truth for program body

  1. x = 8.351211544149733e+102
  2. n = 4.308249766459817e-192

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.147233419817972 \cdot 10^{-08}:\\ \;\;\;\;\left(\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} + \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right) \cdot \left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right)\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.220885236034127 \cdot 10^{-08}:\\ \;\;\;\;\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{\frac{\log x}{n \cdot n}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^{3} - {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}}{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r3233957 = x;
        double r3233958 = 1.0;
        double r3233959 = r3233957 + r3233958;
        double r3233960 = n;
        double r3233961 = r3233958 / r3233960;
        double r3233962 = pow(r3233959, r3233961);
        double r3233963 = pow(r3233957, r3233961);
        double r3233964 = r3233962 - r3233963;
        return r3233964;
}

↓

double f(double x, double n) {
        double r3233965 = 1.0;
        double r3233966 = n;
        double r3233967 = r3233965 / r3233966;
        double r3233968 = -3.147233419817972e-08;
        bool r3233969 = r3233967 <= r3233968;
        double r3233970 = x;
        double r3233971 = r3233970 + r3233965;
        double r3233972 = pow(r3233971, r3233967);
        double r3233973 = sqrt(r3233972);
        double r3233974 = 2.0;
        double r3233975 = r3233967 / r3233974;
        double r3233976 = pow(r3233970, r3233975);
        double r3233977 = r3233973 + r3233976;
        double r3233978 = sqrt(r3233973);
        double r3233979 = sqrt(r3233976);
        double r3233980 = r3233978 + r3233979;
        double r3233981 = r3233978 - r3233979;
        double r3233982 = r3233980 * r3233981;
        double r3233983 = r3233977 * r3233982;
        double r3233984 = cbrt(r3233983);
        double r3233985 = pow(r3233970, r3233967);
        double r3233986 = r3233972 - r3233985;
        double r3233987 = cbrt(r3233986);
        double r3233988 = r3233984 * r3233987;
        double r3233989 = r3233973 - r3233976;
        double r3233990 = r3233989 * r3233977;
        double r3233991 = cbrt(r3233990);
        double r3233992 = r3233988 * r3233991;
        double r3233993 = 2.220885236034127e-08;
        bool r3233994 = r3233967 <= r3233993;
        double r3233995 = r3233970 * r3233966;
        double r3233996 = r3233965 / r3233995;
        double r3233997 = 0.5;
        double r3233998 = r3233997 / r3233966;
        double r3233999 = r3233970 * r3233970;
        double r3234000 = r3233998 / r3233999;
        double r3234001 = log(r3233970);
        double r3234002 = r3233966 * r3233966;
        double r3234003 = r3234001 / r3234002;
        double r3234004 = r3234003 / r3233970;
        double r3234005 = r3234000 - r3234004;
        double r3234006 = r3233996 - r3234005;
        double r3234007 = 3.0;
        double r3234008 = pow(r3233973, r3234007);
        double r3234009 = pow(r3233976, r3234007);
        double r3234010 = r3234008 - r3234009;
        double r3234011 = r3233973 * r3233973;
        double r3234012 = r3233976 * r3233976;
        double r3234013 = r3233976 * r3233973;
        double r3234014 = r3234012 + r3234013;
        double r3234015 = r3234011 + r3234014;
        double r3234016 = r3234010 / r3234015;
        double r3234017 = r3234016 * r3233977;
        double r3234018 = cbrt(r3234017);
        double r3234019 = r3233987 * r3233987;
        double r3234020 = r3234018 * r3234019;
        double r3234021 = r3233994 ? r3234006 : r3234020;
        double r3234022 = r3233969 ? r3233992 : r3234021;
        return r3234022;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1 n) < -3.147233419817972e-08

   1. Initial program 0.8

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

   3. Applied add-cube-cbrt 0.8

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

   5. Applied sqr-pow 0.8

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

   6. Applied add-sqr-sqrt 0.8

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

   7. Applied difference-of-squares 0.8

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

   9. Applied sqr-pow 0.8

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

   10. Applied add-sqr-sqrt 0.8

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

   11. Applied difference-of-squares 0.8

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

   13. Applied add-sqr-sqrt 0.8

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

   14. Applied add-sqr-sqrt 0.8

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

   15. Applied sqrt-prod 0.8

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

   16. Applied difference-of-squares 0.8

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

   if -3.147233419817972e-08 < (/ 1 n) < 2.220885236034127e-08

   1. Initial program 45.2

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

   2. Taylor expanded around inf 32.6

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

   3. Simplified 32.5

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

   if 2.220885236034127e-08 < (/ 1 n)

   1. Initial program 25.7

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

   3. Applied add-cube-cbrt 25.7

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

   5. Applied sqr-pow 25.7

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

   6. Applied add-sqr-sqrt 25.7

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

   7. Applied difference-of-squares 25.7

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

   9. Applied flip3-- 28.7

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

4. Final simplification 22.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.147233419817972 \cdot 10^{-08}:\\ \;\;\;\;\left(\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} + \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right) \cdot \left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right)\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.220885236034127 \cdot 10^{-08}:\\ \;\;\;\;\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{\frac{\log x}{n \cdot n}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^{3} - {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}}{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))