2nthrt (problem 3.4.6)

Average Error: 29.2 → 18.8

Time: 44.4s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.3097472283775323e+293
  2. n = 3.7854182562615194e-287

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -6473591223290.638:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{-\frac{\log x}{x}}{n \cdot n}\right) - \frac{\frac{1}{2}}{\left(x \cdot n\right) \cdot x}\\ \mathbf{elif}\;n \le 8.2787868631613 \cdot 10^{-311}:\\ \;\;\;\;\sqrt[3]{\log \left(e^{\mathsf{fma}\left(\left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right), \left(\left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)\right)}\right)} \cdot \left(\sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{\mathsf{fma}\left(\left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right), \left(-{x}^{\left(\frac{1}{n}\right)}\right)\right)}\right)}\right)\\ \mathbf{elif}\;n \le 222183268.51354733:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{-\frac{\log x}{x}}{n \cdot n}\right) - \frac{\frac{1}{2}}{\left(x \cdot n\right) \cdot x}\\ \end{array}\]

C

double f(double x, double n) {
        double r1980978 = x;
        double r1980979 = 1.0;
        double r1980980 = r1980978 + r1980979;
        double r1980981 = n;
        double r1980982 = r1980979 / r1980981;
        double r1980983 = pow(r1980980, r1980982);
        double r1980984 = pow(r1980978, r1980982);
        double r1980985 = r1980983 - r1980984;
        return r1980985;
}

↓

double f(double x, double n) {
        double r1980986 = n;
        double r1980987 = -6473591223290.638;
        bool r1980988 = r1980986 <= r1980987;
        double r1980989 = 1.0;
        double r1980990 = r1980989 / r1980986;
        double r1980991 = x;
        double r1980992 = r1980990 / r1980991;
        double r1980993 = log(r1980991);
        double r1980994 = r1980993 / r1980991;
        double r1980995 = -r1980994;
        double r1980996 = r1980986 * r1980986;
        double r1980997 = r1980995 / r1980996;
        double r1980998 = r1980992 - r1980997;
        double r1980999 = 0.5;
        double r1981000 = r1980991 * r1980986;
        double r1981001 = r1981000 * r1980991;
        double r1981002 = r1980999 / r1981001;
        double r1981003 = r1980998 - r1981002;
        double r1981004 = 8.2787868631613e-311;
        bool r1981005 = r1980986 <= r1981004;
        double r1981006 = r1980989 + r1980991;
        double r1981007 = pow(r1981006, r1980990);
        double r1981008 = cbrt(r1981007);
        double r1981009 = r1981008 * r1981008;
        double r1981010 = pow(r1980991, r1980990);
        double r1981011 = cbrt(r1981010);
        double r1981012 = -r1981011;
        double r1981013 = r1981011 * r1981011;
        double r1981014 = r1981012 * r1981013;
        double r1981015 = fma(r1981009, r1981008, r1981014);
        double r1981016 = exp(r1981015);
        double r1981017 = log(r1981016);
        double r1981018 = cbrt(r1981017);
        double r1981019 = r1981007 - r1981010;
        double r1981020 = exp(r1981019);
        double r1981021 = log(r1981020);
        double r1981022 = cbrt(r1981021);
        double r1981023 = -r1981010;
        double r1981024 = fma(r1981009, r1981008, r1981023);
        double r1981025 = exp(r1981024);
        double r1981026 = log(r1981025);
        double r1981027 = cbrt(r1981026);
        double r1981028 = r1981022 * r1981027;
        double r1981029 = r1981018 * r1981028;
        double r1981030 = 222183268.51354733;
        bool r1981031 = r1980986 <= r1981030;
        double r1981032 = log1p(r1980991);
        double r1981033 = r1981032 / r1980986;
        double r1981034 = exp(r1981033);
        double r1981035 = r1981034 - r1981010;
        double r1981036 = r1981031 ? r1981035 : r1981003;
        double r1981037 = r1981005 ? r1981029 : r1981036;
        double r1981038 = r1980988 ? r1981003 : r1981037;
        return r1981038;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if n < -6473591223290.638 or 222183268.51354733 < n

   1. Initial program 45.0

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

   3. Applied add-log-exp 45.0

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

   4. Taylor expanded around inf 32.4

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

   5. Simplified 31.8

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

   if -6473591223290.638 < n < 8.2787868631613e-311

   1. Initial program 1.3

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

   3. Applied add-log-exp 1.4

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

   5. Applied add-cube-cbrt 1.4

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

   7. Applied *-un-lft-identity 1.4

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

   8. Applied add-cube-cbrt 1.4

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

   9. Applied prod-diff 1.4

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

   10. Applied exp-sum 1.4

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

   11. Applied log-prod 1.4

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

   12. Simplified 1.4

       \[\leadsto \left(\sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{\mathsf{fma}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(-{x}^{\left(\frac{1}{n}\right)} \cdot 1\right)\right)}\right) + \color{blue}{0}}\right) \cdot \sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]
   13. Using strategy rm

   14. Applied add-cube-cbrt 1.4

       \[\leadsto \left(\sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{\mathsf{fma}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(-{x}^{\left(\frac{1}{n}\right)} \cdot 1\right)\right)}\right) + 0}\right) \cdot \sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}}}\right)}\]

   15. Applied add-cube-cbrt 1.4

       \[\leadsto \left(\sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{\mathsf{fma}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(-{x}^{\left(\frac{1}{n}\right)} \cdot 1\right)\right)}\right) + 0}\right) \cdot \sqrt[3]{\log \left(e^{\color{blue}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]

   16. Applied prod-diff 1.4

       \[\leadsto \left(\sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{\mathsf{fma}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(-{x}^{\left(\frac{1}{n}\right)} \cdot 1\right)\right)}\right) + 0}\right) \cdot \sqrt[3]{\log \left(e^{\color{blue}{\mathsf{fma}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)\right) + \mathsf{fma}\left(\left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)\right)}}\right)}\]

   17. Applied exp-sum 1.4

       \[\leadsto \left(\sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{\mathsf{fma}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(-{x}^{\left(\frac{1}{n}\right)} \cdot 1\right)\right)}\right) + 0}\right) \cdot \sqrt[3]{\log \color{blue}{\left(e^{\mathsf{fma}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)\right)} \cdot e^{\mathsf{fma}\left(\left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)\right)}\right)}}\]

   18. Applied log-prod 1.4

       \[\leadsto \left(\sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{\mathsf{fma}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(-{x}^{\left(\frac{1}{n}\right)} \cdot 1\right)\right)}\right) + 0}\right) \cdot \sqrt[3]{\color{blue}{\log \left(e^{\mathsf{fma}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)\right)}\right) + \log \left(e^{\mathsf{fma}\left(\left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)\right)}\right)}}\]

   19. Simplified 1.4

       \[\leadsto \left(\sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{\mathsf{fma}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(-{x}^{\left(\frac{1}{n}\right)} \cdot 1\right)\right)}\right) + 0}\right) \cdot \sqrt[3]{\log \left(e^{\mathsf{fma}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right), \left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)\right)}\right) + \color{blue}{0}}\]

   if 8.2787868631613e-311 < n < 222183268.51354733

   1. Initial program 23.6

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

   3. Applied pow-to-exp 23.7

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

   4. Simplified 2.4

      \[\leadsto e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 18.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -6473591223290.638:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{-\frac{\log x}{x}}{n \cdot n}\right) - \frac{\frac{1}{2}}{\left(x \cdot n\right) \cdot x}\\ \mathbf{elif}\;n \le 8.2787868631613 \cdot 10^{-311}:\\ \;\;\;\;\sqrt[3]{\log \left(e^{\mathsf{fma}\left(\left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right), \left(\left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)\right)}\right)} \cdot \left(\sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{\mathsf{fma}\left(\left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right), \left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right), \left(-{x}^{\left(\frac{1}{n}\right)}\right)\right)}\right)}\right)\\ \mathbf{elif}\;n \le 222183268.51354733:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{-\frac{\log x}{x}}{n \cdot n}\right) - \frac{\frac{1}{2}}{\left(x \cdot n\right) \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))