2nthrt (problem 3.4.6)

Average Error: 29.3 → 22.1

Time: 38.9s

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}:\\ \;\;\;\;\sqrt[3]{\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[3]{\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}} \cdot \left(\sqrt[3]{\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}}\right)\right)\right) \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{\left(x + 1\right)}^{\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)\\ \mathbf{elif}\;\frac{1}{n} \le 2.610123296819638228104037622201438750835 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(\frac{-\frac{\log x}{n \cdot n}}{x}, 1, \frac{\frac{0.5}{n}}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\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 \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}\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) + \left({\left(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)}\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r3195671 = x;
        double r3195672 = 1.0;
        double r3195673 = r3195671 + r3195672;
        double r3195674 = n;
        double r3195675 = r3195672 / r3195674;
        double r3195676 = pow(r3195673, r3195675);
        double r3195677 = pow(r3195671, r3195675);
        double r3195678 = r3195676 - r3195677;
        return r3195678;
}

↓

double f(double x, double n) {
        double r3195679 = 1.0;
        double r3195680 = n;
        double r3195681 = r3195679 / r3195680;
        double r3195682 = -1.1987118596397398e-14;
        bool r3195683 = r3195681 <= r3195682;
        double r3195684 = x;
        double r3195685 = r3195684 + r3195679;
        double r3195686 = pow(r3195685, r3195681);
        double r3195687 = sqrt(r3195686);
        double r3195688 = sqrt(r3195687);
        double r3195689 = 2.0;
        double r3195690 = r3195681 / r3195689;
        double r3195691 = pow(r3195684, r3195690);
        double r3195692 = sqrt(r3195691);
        double r3195693 = r3195688 + r3195692;
        double r3195694 = r3195688 - r3195692;
        double r3195695 = cbrt(r3195694);
        double r3195696 = r3195695 * r3195695;
        double r3195697 = r3195695 * r3195696;
        double r3195698 = r3195693 * r3195697;
        double r3195699 = r3195691 + r3195687;
        double r3195700 = r3195698 * r3195699;
        double r3195701 = cbrt(r3195700);
        double r3195702 = pow(r3195684, r3195681);
        double r3195703 = r3195686 - r3195702;
        double r3195704 = cbrt(r3195703);
        double r3195705 = r3195704 * r3195704;
        double r3195706 = r3195701 * r3195705;
        double r3195707 = 2.6101232968196382e-06;
        bool r3195708 = r3195681 <= r3195707;
        double r3195709 = r3195679 / r3195684;
        double r3195710 = r3195709 / r3195680;
        double r3195711 = log(r3195684);
        double r3195712 = r3195680 * r3195680;
        double r3195713 = r3195711 / r3195712;
        double r3195714 = -r3195713;
        double r3195715 = r3195714 / r3195684;
        double r3195716 = 0.5;
        double r3195717 = r3195716 / r3195680;
        double r3195718 = r3195684 * r3195684;
        double r3195719 = r3195717 / r3195718;
        double r3195720 = fma(r3195715, r3195679, r3195719);
        double r3195721 = r3195710 - r3195720;
        double r3195722 = r3195693 * r3195694;
        double r3195723 = r3195699 * r3195722;
        double r3195724 = cbrt(r3195723);
        double r3195725 = cbrt(r3195684);
        double r3195726 = pow(r3195725, r3195681);
        double r3195727 = -r3195726;
        double r3195728 = r3195725 * r3195725;
        double r3195729 = pow(r3195728, r3195681);
        double r3195730 = r3195726 * r3195729;
        double r3195731 = fma(r3195727, r3195729, r3195730);
        double r3195732 = r3195686 - r3195730;
        double r3195733 = r3195731 + r3195732;
        double r3195734 = cbrt(r3195733);
        double r3195735 = r3195704 * r3195734;
        double r3195736 = r3195724 * r3195735;
        double r3195737 = r3195708 ? r3195721 : r3195736;
        double r3195738 = r3195683 ? r3195706 : r3195737;
        return r3195738;
}

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 sqr-pow 1.7

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

   6. Applied add-sqr-sqrt 1.7

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

   7. Applied difference-of-squares 1.7

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

   9. Applied add-sqr-sqrt 1.7

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

   10. Applied add-sqr-sqrt 1.7

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

   11. Applied sqrt-prod 1.7

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

   12. Applied difference-of-squares 1.7

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

   14. Applied add-cube-cbrt 1.7

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

   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

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

   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 sqr-pow 24.6

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

   6. Applied add-sqr-sqrt 24.6

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

   7. Applied difference-of-squares 24.6

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

   9. Applied add-sqr-sqrt 24.6

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

   10. Applied add-sqr-sqrt 24.6

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

   11. Applied sqrt-prod 24.6

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

   12. Applied difference-of-squares 24.6

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

   14. Applied add-cube-cbrt 24.6

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

   15. Applied unpow-prod-down 24.6

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

   16. Applied *-un-lft-identity 24.6

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

   17. Applied unpow-prod-down 24.6

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

   18. Applied prod-diff 24.6

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

   19. Simplified 24.6

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

4. Final simplification 22.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.1987118596397397750669009996408489051 \cdot 10^{-14}:\\ \;\;\;\;\sqrt[3]{\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[3]{\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}} \cdot \left(\sqrt[3]{\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{{x}^{\left(\frac{\frac{1}{n}}{2}\right)}}}\right)\right)\right) \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{\left(x + 1\right)}^{\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)\\ \mathbf{elif}\;\frac{1}{n} \le 2.610123296819638228104037622201438750835 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(\frac{-\frac{\log x}{n \cdot n}}{x}, 1, \frac{\frac{0.5}{n}}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\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 \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}\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) + \left({\left(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)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))