2nthrt (problem 3.4.6)

Average Error: 29.4 → 23.2

Time: 1.1m

Precision: 64

• could not determine a ground truth for program body

  1. x = 9.137530169746875e+200
  2. n = 1.4011542472758516e-71

Math

\[{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1.0}{n} \le -6.88847562666282 \cdot 10^{-20}:\\ \;\;\;\;\sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}} \cdot \left(\sqrt[3]{\log \left(\frac{e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}}}{e^{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}}\right)} \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1.0}{n} \le 1.8205393655820836 \cdot 10^{-32}:\\ \;\;\;\;\left(\frac{\log x \cdot 1.0}{x \cdot \left(n \cdot n\right)} + \frac{\frac{1.0}{n}}{x}\right) - \frac{\frac{0.5}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}\\ \end{array}\]

C

double f(double x, double n) {
        double r2944238 = x;
        double r2944239 = 1.0;
        double r2944240 = r2944238 + r2944239;
        double r2944241 = n;
        double r2944242 = r2944239 / r2944241;
        double r2944243 = pow(r2944240, r2944242);
        double r2944244 = pow(r2944238, r2944242);
        double r2944245 = r2944243 - r2944244;
        return r2944245;
}

↓

double f(double x, double n) {
        double r2944246 = 1.0;
        double r2944247 = n;
        double r2944248 = r2944246 / r2944247;
        double r2944249 = -6.88847562666282e-20;
        bool r2944250 = r2944248 <= r2944249;
        double r2944251 = x;
        double r2944252 = r2944251 + r2944246;
        double r2944253 = pow(r2944252, r2944248);
        double r2944254 = cbrt(r2944251);
        double r2944255 = r2944254 * r2944254;
        double r2944256 = pow(r2944255, r2944248);
        double r2944257 = pow(r2944254, r2944248);
        double r2944258 = r2944256 * r2944257;
        double r2944259 = r2944253 - r2944258;
        double r2944260 = cbrt(r2944259);
        double r2944261 = exp(r2944253);
        double r2944262 = exp(r2944258);
        double r2944263 = r2944261 / r2944262;
        double r2944264 = log(r2944263);
        double r2944265 = cbrt(r2944264);
        double r2944266 = r2944265 * r2944260;
        double r2944267 = r2944260 * r2944266;
        double r2944268 = 1.8205393655820836e-32;
        bool r2944269 = r2944248 <= r2944268;
        double r2944270 = log(r2944251);
        double r2944271 = r2944270 * r2944246;
        double r2944272 = r2944247 * r2944247;
        double r2944273 = r2944251 * r2944272;
        double r2944274 = r2944271 / r2944273;
        double r2944275 = r2944248 / r2944251;
        double r2944276 = r2944274 + r2944275;
        double r2944277 = 0.5;
        double r2944278 = r2944277 / r2944247;
        double r2944279 = r2944251 * r2944251;
        double r2944280 = r2944278 / r2944279;
        double r2944281 = r2944276 - r2944280;
        double r2944282 = r2944260 * r2944260;
        double r2944283 = r2944282 * r2944260;
        double r2944284 = r2944269 ? r2944281 : r2944283;
        double r2944285 = r2944250 ? r2944267 : r2944284;
        return r2944285;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -6.88847562666282e-20

   1. Initial program 2.4

      \[{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 2.4

      \[\leadsto {\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{\left(\frac{1.0}{n}\right)}\]

   4. Applied unpow-prod-down 2.4

      \[\leadsto {\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - \color{blue}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}\]

   5. Applied *-un-lft-identity 2.4

      \[\leadsto {\color{blue}{\left(1 \cdot \left(x + 1.0\right)\right)}}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}\]

   6. Applied unpow-prod-down 2.4

      \[\leadsto \color{blue}{{1}^{\left(\frac{1.0}{n}\right)} \cdot {\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}\]

   7. Applied prod-diff 2.4

      \[\leadsto \color{blue}{\mathsf{fma}\left({1}^{\left(\frac{1.0}{n}\right)}, {\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}, -{\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}\right) + \mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}\right)}\]

   8. Simplified 2.4

      \[\leadsto \color{blue}{\left({\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}\right)} + \mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}\right)\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 2.4

       \[\leadsto \color{blue}{\left(\sqrt[3]{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}} \cdot \sqrt[3]{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}} + \mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}\right)\]

   11. Taylor expanded around 0 2.4

       \[\leadsto \left(\sqrt[3]{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}} \cdot \sqrt[3]{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}} + \color{blue}{0}\]
   12. Using strategy rm

   13. Applied add-log-exp 2.6

       \[\leadsto \left(\sqrt[3]{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - \color{blue}{\log \left(e^{{\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}\right)}} \cdot \sqrt[3]{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}} + 0\]

   14. Applied add-log-exp 2.6

       \[\leadsto \left(\sqrt[3]{\color{blue}{\log \left(e^{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)}}\right)} - \log \left(e^{{\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}\right)} \cdot \sqrt[3]{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}} + 0\]

   15. Applied diff-log 2.6

       \[\leadsto \left(\sqrt[3]{\color{blue}{\log \left(\frac{e^{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)}}}{e^{{\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}}\right)}} \cdot \sqrt[3]{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}} + 0\]

   if -6.88847562666282e-20 < (/ 1.0 n) < 1.8205393655820836e-32

   1. Initial program 44.1

      \[{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}\]

   2. Taylor expanded around inf 33.1

      \[\leadsto \color{blue}{1.0 \cdot \frac{1}{x \cdot n} - \left(1.0 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}} + 0.5 \cdot \frac{1}{{x}^{2} \cdot n}\right)}\]

   3. Simplified 32.5

      \[\leadsto \color{blue}{\left(\frac{\frac{1.0}{n}}{x} + \frac{\log x \cdot 1.0}{x \cdot \left(n \cdot n\right)}\right) - \frac{\frac{0.5}{n}}{x \cdot x}}\]

   if 1.8205393655820836e-32 < (/ 1.0 n)

   1. Initial program 29.2

      \[{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 29.2

      \[\leadsto {\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{\left(\frac{1.0}{n}\right)}\]

   4. Applied unpow-prod-down 29.2

      \[\leadsto {\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - \color{blue}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}\]

   5. Applied *-un-lft-identity 29.2

      \[\leadsto {\color{blue}{\left(1 \cdot \left(x + 1.0\right)\right)}}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}\]

   6. Applied unpow-prod-down 29.2

      \[\leadsto \color{blue}{{1}^{\left(\frac{1.0}{n}\right)} \cdot {\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}\]

   7. Applied prod-diff 29.2

      \[\leadsto \color{blue}{\mathsf{fma}\left({1}^{\left(\frac{1.0}{n}\right)}, {\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}, -{\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}\right) + \mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}\right)}\]

   8. Simplified 29.2

      \[\leadsto \color{blue}{\left({\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}\right)} + \mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}\right)\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 29.3

       \[\leadsto \color{blue}{\left(\sqrt[3]{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}} \cdot \sqrt[3]{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}} + \mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}\right)\]

   11. Taylor expanded around 0 29.2

       \[\leadsto \left(\sqrt[3]{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}} \cdot \sqrt[3]{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(1.0 + x\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}} + \color{blue}{0}\]
3. Recombined 3 regimes into one program.

4. Final simplification 23.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1.0}{n} \le -6.88847562666282 \cdot 10^{-20}:\\ \;\;\;\;\sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}} \cdot \left(\sqrt[3]{\log \left(\frac{e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}}}{e^{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}}\right)} \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1.0}{n} \le 1.8205393655820836 \cdot 10^{-32}:\\ \;\;\;\;\left(\frac{\log x \cdot 1.0}{x \cdot \left(n \cdot n\right)} + \frac{\frac{1.0}{n}}{x}\right) - \frac{\frac{0.5}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1.0}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))