Average Error: 28.9 → 21.8
Time: 29.7s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -29869477.165418986:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{1}{x \cdot n} \cdot \frac{\log x}{n}\right) - \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot n}\\ \mathbf{elif}\;n \le 0.9292141684675859:\\ \;\;\;\;\left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n} \cdot \frac{1}{3}\right)} \cdot {\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{1}{x \cdot n} \cdot \frac{\log x}{n}\right) - \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot n}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -29869477.165418986:\\
\;\;\;\;\left(\frac{1}{x \cdot n} + \frac{1}{x \cdot n} \cdot \frac{\log x}{n}\right) - \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot n}\\

\mathbf{elif}\;n \le 0.9292141684675859:\\
\;\;\;\;\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) \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n} \cdot \frac{1}{3}\right)} \cdot {\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{x \cdot n} + \frac{1}{x \cdot n} \cdot \frac{\log x}{n}\right) - \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot n}\\

\end{array}
double f(double x, double n) {
        double r1262624 = x;
        double r1262625 = 1.0;
        double r1262626 = r1262624 + r1262625;
        double r1262627 = n;
        double r1262628 = r1262625 / r1262627;
        double r1262629 = pow(r1262626, r1262628);
        double r1262630 = pow(r1262624, r1262628);
        double r1262631 = r1262629 - r1262630;
        return r1262631;
}
↓
double f(double x, double n) {
        double r1262632 = n;
        double r1262633 = -29869477.165418986;
        bool r1262634 = r1262632 <= r1262633;
        double r1262635 = 1.0;
        double r1262636 = x;
        double r1262637 = r1262636 * r1262632;
        double r1262638 = r1262635 / r1262637;
        double r1262639 = log(r1262636);
        double r1262640 = r1262639 / r1262632;
        double r1262641 = r1262638 * r1262640;
        double r1262642 = r1262638 + r1262641;
        double r1262643 = 0.5;
        double r1262644 = r1262636 * r1262636;
        double r1262645 = r1262644 * r1262632;
        double r1262646 = r1262643 / r1262645;
        double r1262647 = r1262642 - r1262646;
        double r1262648 = 0.9292141684675859;
        bool r1262649 = r1262632 <= r1262648;
        double r1262650 = r1262635 + r1262636;
        double r1262651 = r1262635 / r1262632;
        double r1262652 = pow(r1262650, r1262651);
        double r1262653 = cbrt(r1262652);
        double r1262654 = r1262653 * r1262653;
        double r1262655 = 0.3333333333333333;
        double r1262656 = r1262651 * r1262655;
        double r1262657 = pow(r1262650, r1262656);
        double r1262658 = cbrt(r1262650);
        double r1262659 = r1262658 * r1262658;
        double r1262660 = pow(r1262659, r1262651);
        double r1262661 = r1262657 * r1262660;
        double r1262662 = cbrt(r1262661);
        double r1262663 = r1262654 * r1262662;
        double r1262664 = pow(r1262636, r1262651);
        double r1262665 = r1262663 - r1262664;
        double r1262666 = r1262649 ? r1262665 : r1262647;
        double r1262667 = r1262634 ? r1262647 : r1262666;
        return r1262667;
}

Error

Bits error versus x

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if n < -29869477.165418986 or 0.9292141684675859 < n

    1. Initial program 44.4

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

    2. Taylor expanded around inf 32.1

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

    3. Simplified32.1

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

    if -29869477.165418986 < n < 0.9292141684675859

    1. Initial program 7.9

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

    3. Applied add-cube-cbrt7.9

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

    5. Applied add-cube-cbrt7.9

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

    6. Applied unpow-prod-down7.9

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

    8. Applied pow1/328.7

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

    9. Applied pow-pow7.9

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

  4. Final simplification21.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -29869477.165418986:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{1}{x \cdot n} \cdot \frac{\log x}{n}\right) - \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot n}\\ \mathbf{elif}\;n \le 0.9292141684675859:\\ \;\;\;\;\left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n} \cdot \frac{1}{3}\right)} \cdot {\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{1}{x \cdot n} \cdot \frac{\log x}{n}\right) - \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))