Average Error: 29.2 → 21.7
Time: 30.2s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -6473591223290.638:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{\frac{1}{2}}{x}}{x \cdot n} - \frac{1}{n} \cdot \frac{\log x}{x \cdot n}\right)\\ \mathbf{elif}\;n \le 222183268.51354733:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{\frac{1}{2}}{x}}{x \cdot n} - \frac{1}{n} \cdot \frac{\log x}{x \cdot n}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -6473591223290.638:\\
\;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{\frac{1}{2}}{x}}{x \cdot n} - \frac{1}{n} \cdot \frac{\log x}{x \cdot n}\right)\\

\mathbf{elif}\;n \le 222183268.51354733:\\
\;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \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)\\

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

\end{array}
double f(double x, double n) {
        double r2447044 = x;
        double r2447045 = 1.0;
        double r2447046 = r2447044 + r2447045;
        double r2447047 = n;
        double r2447048 = r2447045 / r2447047;
        double r2447049 = pow(r2447046, r2447048);
        double r2447050 = pow(r2447044, r2447048);
        double r2447051 = r2447049 - r2447050;
        return r2447051;
}
↓
double f(double x, double n) {
        double r2447052 = n;
        double r2447053 = -6473591223290.638;
        bool r2447054 = r2447052 <= r2447053;
        double r2447055 = 1.0;
        double r2447056 = r2447055 / r2447052;
        double r2447057 = x;
        double r2447058 = r2447056 / r2447057;
        double r2447059 = 0.5;
        double r2447060 = r2447059 / r2447057;
        double r2447061 = r2447057 * r2447052;
        double r2447062 = r2447060 / r2447061;
        double r2447063 = log(r2447057);
        double r2447064 = r2447063 / r2447061;
        double r2447065 = r2447056 * r2447064;
        double r2447066 = r2447062 - r2447065;
        double r2447067 = r2447058 - r2447066;
        double r2447068 = 222183268.51354733;
        bool r2447069 = r2447052 <= r2447068;
        double r2447070 = r2447055 + r2447057;
        double r2447071 = pow(r2447070, r2447056);
        double r2447072 = pow(r2447057, r2447056);
        double r2447073 = cbrt(r2447072);
        double r2447074 = r2447073 * r2447073;
        double r2447075 = r2447073 * r2447074;
        double r2447076 = r2447071 - r2447075;
        double r2447077 = r2447069 ? r2447076 : r2447067;
        double r2447078 = r2447054 ? r2447067 : r2447077;
        return r2447078;
}

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 < -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-cube-cbrt45.1

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

    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. Simplified31.7

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

    if -6473591223290.638 < n < 222183268.51354733

    1. Initial program 8.2

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

    3. Applied add-cube-cbrt8.3

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

  4. Final simplification21.7

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

Reproduce

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