2nthrt (problem 3.4.6)

Average Error: 29.3 → 22.6

Time: 29.2s

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)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} + \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \left(\sqrt[3]{\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}} \cdot \left(\sqrt[3]{\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}} \cdot \sqrt[3]{\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}}\right)\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.610123296819638228104037622201438750835 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} + \frac{1 \cdot \log x}{x \cdot \left(n \cdot n\right)}\right) - \frac{\frac{0.5}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\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{1}{n}\right)}} \cdot \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}}{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)} \cdot \left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} + \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\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 r2620092 = x;
        double r2620093 = 1.0;
        double r2620094 = r2620092 + r2620093;
        double r2620095 = n;
        double r2620096 = r2620093 / r2620095;
        double r2620097 = pow(r2620094, r2620096);
        double r2620098 = pow(r2620092, r2620096);
        double r2620099 = r2620097 - r2620098;
        return r2620099;
}

↓

double f(double x, double n) {
        double r2620100 = 1.0;
        double r2620101 = n;
        double r2620102 = r2620100 / r2620101;
        double r2620103 = -1.1987118596397398e-14;
        bool r2620104 = r2620102 <= r2620103;
        double r2620105 = x;
        double r2620106 = r2620105 + r2620100;
        double r2620107 = pow(r2620106, r2620102);
        double r2620108 = pow(r2620105, r2620102);
        double r2620109 = r2620107 - r2620108;
        double r2620110 = cbrt(r2620109);
        double r2620111 = r2620110 * r2620110;
        double r2620112 = sqrt(r2620107);
        double r2620113 = sqrt(r2620108);
        double r2620114 = r2620112 + r2620113;
        double r2620115 = sqrt(r2620112);
        double r2620116 = sqrt(r2620113);
        double r2620117 = r2620115 + r2620116;
        double r2620118 = r2620115 - r2620116;
        double r2620119 = cbrt(r2620118);
        double r2620120 = r2620119 * r2620119;
        double r2620121 = r2620119 * r2620120;
        double r2620122 = r2620117 * r2620121;
        double r2620123 = r2620114 * r2620122;
        double r2620124 = cbrt(r2620123);
        double r2620125 = r2620111 * r2620124;
        double r2620126 = 2.6101232968196382e-06;
        bool r2620127 = r2620102 <= r2620126;
        double r2620128 = r2620100 / r2620105;
        double r2620129 = r2620128 / r2620101;
        double r2620130 = log(r2620105);
        double r2620131 = r2620100 * r2620130;
        double r2620132 = r2620101 * r2620101;
        double r2620133 = r2620105 * r2620132;
        double r2620134 = r2620131 / r2620133;
        double r2620135 = r2620129 + r2620134;
        double r2620136 = 0.5;
        double r2620137 = r2620136 / r2620101;
        double r2620138 = r2620105 * r2620105;
        double r2620139 = r2620137 / r2620138;
        double r2620140 = r2620135 - r2620139;
        double r2620141 = r2620115 * r2620112;
        double r2620142 = r2620113 * r2620116;
        double r2620143 = r2620141 - r2620142;
        double r2620144 = r2620115 * r2620116;
        double r2620145 = r2620113 + r2620144;
        double r2620146 = r2620112 + r2620145;
        double r2620147 = r2620143 / r2620146;
        double r2620148 = r2620147 * r2620117;
        double r2620149 = r2620148 * r2620114;
        double r2620150 = cbrt(r2620149);
        double r2620151 = r2620150 * r2620111;
        double r2620152 = r2620127 ? r2620140 : r2620151;
        double r2620153 = r2620104 ? r2620125 : r2620152;
        return r2620153;
}

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 add-sqr-sqrt 1.7

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

   6. Applied add-sqr-sqrt 1.7

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

   7. Applied difference-of-squares 1.7

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

   9. Applied add-sqr-sqrt 1.7

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

   10. Applied sqrt-prod 1.7

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

   11. Applied add-sqr-sqrt 1.7

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

   12. Applied sqrt-prod 1.7

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

   13. Applied difference-of-squares 1.7

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

   15. Applied add-cube-cbrt 1.7

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

   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 31.7

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

   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 add-sqr-sqrt 24.6

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

   6. Applied add-sqr-sqrt 24.6

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

   7. Applied difference-of-squares 24.6

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

   9. Applied add-sqr-sqrt 24.6

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

   10. Applied sqrt-prod 24.6

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

   11. Applied add-sqr-sqrt 24.6

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

   12. Applied sqrt-prod 24.6

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

   13. Applied difference-of-squares 24.6

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

   15. Applied flip3-- 28.0

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

   16. Simplified 28.0

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

   17. Simplified 28.0

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

4. Final simplification 22.6

   \[\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)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} + \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \left(\sqrt[3]{\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}} \cdot \left(\sqrt[3]{\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}} \cdot \sqrt[3]{\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}}\right)\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.610123296819638228104037622201438750835 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} + \frac{1 \cdot \log x}{x \cdot \left(n \cdot n\right)}\right) - \frac{\frac{0.5}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\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{1}{n}\right)}} \cdot \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}}{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)} \cdot \left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} + \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\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 2019174 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))