2nthrt (problem 3.4.6)

Average Error: 33.0 → 24.4

Time: 18.5s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.199947025871061e+147
  2. n = 1.4433187213430084e-221

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -7.7456503819782359 \cdot 10^{-30}:\\ \;\;\;\;\left(\sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right) + \mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 3.6323271639935607 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\mathsf{fma}\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

C

double f(double x, double n) {
        double r100959 = x;
        double r100960 = 1.0;
        double r100961 = r100959 + r100960;
        double r100962 = n;
        double r100963 = r100960 / r100962;
        double r100964 = pow(r100961, r100963);
        double r100965 = pow(r100959, r100963);
        double r100966 = r100964 - r100965;
        return r100966;
}

↓

double f(double x, double n) {
        double r100967 = 1.0;
        double r100968 = n;
        double r100969 = r100967 / r100968;
        double r100970 = -7.745650381978236e-30;
        bool r100971 = r100969 <= r100970;
        double r100972 = x;
        double r100973 = r100972 + r100967;
        double r100974 = pow(r100973, r100969);
        double r100975 = sqrt(r100972);
        double r100976 = 2.0;
        double r100977 = r100976 * r100969;
        double r100978 = pow(r100975, r100977);
        double r100979 = r100974 - r100978;
        double r100980 = pow(r100975, r100969);
        double r100981 = -r100980;
        double r100982 = fma(r100980, r100981, r100978);
        double r100983 = r100979 + r100982;
        double r100984 = cbrt(r100983);
        double r100985 = pow(r100972, r100969);
        double r100986 = r100974 - r100985;
        double r100987 = cbrt(r100986);
        double r100988 = r100984 * r100987;
        double r100989 = sqrt(r100973);
        double r100990 = pow(r100989, r100969);
        double r100991 = r100969 / r100976;
        double r100992 = pow(r100972, r100991);
        double r100993 = r100990 + r100992;
        double r100994 = r100990 - r100992;
        double r100995 = r100993 * r100994;
        double r100996 = cbrt(r100995);
        double r100997 = r100988 * r100996;
        double r100998 = 3.6323271639935607e-11;
        bool r100999 = r100969 <= r100998;
        double r101000 = 1.0;
        double r101001 = r100972 * r100968;
        double r101002 = r101000 / r101001;
        double r101003 = 0.5;
        double r101004 = pow(r100972, r100976);
        double r101005 = r101004 * r100968;
        double r101006 = r101000 / r101005;
        double r101007 = r101000 / r100972;
        double r101008 = log(r101007);
        double r101009 = pow(r100968, r100976);
        double r101010 = r100972 * r101009;
        double r101011 = r101008 / r101010;
        double r101012 = r100967 * r101011;
        double r101013 = fma(r101003, r101006, r101012);
        double r101014 = -r101013;
        double r101015 = fma(r100967, r101002, r101014);
        double r101016 = cbrt(r100973);
        double r101017 = r101016 * r101016;
        double r101018 = pow(r101017, r100969);
        double r101019 = pow(r101016, r100969);
        double r101020 = cbrt(r100972);
        double r101021 = pow(r101020, r100969);
        double r101022 = r101020 * r101020;
        double r101023 = pow(r101022, r100969);
        double r101024 = r101021 * r101023;
        double r101025 = -r101024;
        double r101026 = fma(r101018, r101019, r101025);
        double r101027 = -r101021;
        double r101028 = r101027 + r101021;
        double r101029 = r101023 * r101028;
        double r101030 = r101026 + r101029;
        double r101031 = cbrt(r101030);
        double r101032 = r100987 * r101031;
        double r101033 = r101032 * r100987;
        double r101034 = r100999 ? r101015 : r101033;
        double r101035 = r100971 ? r100997 : r101034;
        return r101035;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -7.745650381978236e-30

   1. Initial program 7.6

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

   3. Applied add-cube-cbrt 7.6

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

   5. Applied add-sqr-sqrt 7.6

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

   6. Applied unpow-prod-down 7.6

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

   7. Applied *-un-lft-identity 7.6

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

   8. Applied unpow-prod-down 7.6

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

   9. Applied prod-diff 7.6

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

   10. Simplified 7.6

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

   11. Simplified 7.6

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

   13. Applied sqr-pow 7.6

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

   14. Applied add-sqr-sqrt 7.6

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

   15. Applied unpow-prod-down 7.6

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

   16. Applied difference-of-squares 7.6

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

   if -7.745650381978236e-30 < (/ 1.0 n) < 3.6323271639935607e-11

   1. Initial program 44.9

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

   2. Taylor expanded around inf 32.4

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

   3. Simplified 32.4

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

   if 3.6323271639935607e-11 < (/ 1.0 n)

   1. Initial program 7.4

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

   3. Applied add-cube-cbrt 7.4

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

   5. Applied add-cube-cbrt 7.4

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

   6. Applied unpow-prod-down 7.4

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

   7. Applied add-cube-cbrt 7.4

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

   8. Applied unpow-prod-down 7.4

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

   9. Applied prod-diff 7.4

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

   10. Simplified 7.4

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

4. Final simplification 24.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -7.7456503819782359 \cdot 10^{-30}:\\ \;\;\;\;\left(\sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right) + \mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 3.6323271639935607 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\mathsf{fma}\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))