Average Error: 29.4 → 22.3
Time: 31.9s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.5351028006273464976416676108783576637506 \lor \neg \left(\frac{1}{n} \le 4.133628459687642410799750514629715336321 \cdot 10^{-12}\right):\\ \;\;\;\;{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left({x}^{1}\right)}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} + 1 \cdot \frac{\log x}{x \cdot {n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -0.5351028006273464976416676108783576637506 \lor \neg \left(\frac{1}{n} \le 4.133628459687642410799750514629715336321 \cdot 10^{-12}\right):\\
\;\;\;\;{\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)} - {\left({x}^{1}\right)}^{\left(\frac{1}{n}\right)}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\frac{1}{x}}{n} + 1 \cdot \frac{\log x}{x \cdot {n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\

\end{array}
double f(double x, double n) {
        double r71818 = x;
        double r71819 = 1.0;
        double r71820 = r71818 + r71819;
        double r71821 = n;
        double r71822 = r71819 / r71821;
        double r71823 = pow(r71820, r71822);
        double r71824 = pow(r71818, r71822);
        double r71825 = r71823 - r71824;
        return r71825;
}
↓
double f(double x, double n) {
        double r71826 = 1.0;
        double r71827 = n;
        double r71828 = r71826 / r71827;
        double r71829 = -0.5351028006273465;
        bool r71830 = r71828 <= r71829;
        double r71831 = 4.1336284596876424e-12;
        bool r71832 = r71828 <= r71831;
        double r71833 = !r71832;
        bool r71834 = r71830 || r71833;
        double r71835 = x;
        double r71836 = r71835 + r71826;
        double r71837 = cbrt(r71836);
        double r71838 = r71837 * r71837;
        double r71839 = pow(r71838, r71828);
        double r71840 = pow(r71837, r71828);
        double r71841 = r71839 * r71840;
        double r71842 = pow(r71835, r71826);
        double r71843 = 1.0;
        double r71844 = r71843 / r71827;
        double r71845 = pow(r71842, r71844);
        double r71846 = r71841 - r71845;
        double r71847 = r71826 / r71835;
        double r71848 = r71847 / r71827;
        double r71849 = log(r71835);
        double r71850 = 2.0;
        double r71851 = pow(r71827, r71850);
        double r71852 = r71835 * r71851;
        double r71853 = r71849 / r71852;
        double r71854 = r71826 * r71853;
        double r71855 = r71848 + r71854;
        double r71856 = 0.5;
        double r71857 = pow(r71835, r71850);
        double r71858 = r71857 * r71827;
        double r71859 = r71856 / r71858;
        double r71860 = r71855 - r71859;
        double r71861 = r71834 ? r71846 : r71860;
        return r71861;
}

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 (/ 1.0 n) < -0.5351028006273465 or 4.1336284596876424e-12 < (/ 1.0 n)

    1. Initial program 8.5

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

    3. Applied add-cube-cbrt8.6

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

    4. Applied unpow-prod-down8.6

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

    6. Applied div-inv8.6

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

    7. Applied pow-unpow8.6

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

    if -0.5351028006273465 < (/ 1.0 n) < 4.1336284596876424e-12

    1. Initial program 44.6

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

    2. Taylor expanded around inf 32.9

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

    3. Simplified32.3

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}}\]
    4. Using strategy rm

    5. Applied sub-neg32.3

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

    6. Applied distribute-lft-in32.3

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

    7. Simplified32.3

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

    8. Simplified32.3

      \[\leadsto \left(\frac{\frac{1}{x}}{n} + \color{blue}{1 \cdot \frac{\log x}{x \cdot {n}^{2}}}\right) - \frac{0.5}{{x}^{2} \cdot n}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification22.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.5351028006273464976416676108783576637506 \lor \neg \left(\frac{1}{n} \le 4.133628459687642410799750514629715336321 \cdot 10^{-12}\right):\\ \;\;\;\;{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {\left({x}^{1}\right)}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} + 1 \cdot \frac{\log x}{x \cdot {n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))