Average Error: 29.0 → 21.9
Time: 12.5s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -9.91421720864949135 \cdot 10^{-7} \lor \neg \left(\frac{1}{n} \le 1.63989863279117905 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -9.91421720864949135 \cdot 10^{-7} \lor \neg \left(\frac{1}{n} \le 1.63989863279117905 \cdot 10^{-10}\right):\\
\;\;\;\;\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)}}\\

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

\end{array}
double f(double x, double n) {
        double r67746 = x;
        double r67747 = 1.0;
        double r67748 = r67746 + r67747;
        double r67749 = n;
        double r67750 = r67747 / r67749;
        double r67751 = pow(r67748, r67750);
        double r67752 = pow(r67746, r67750);
        double r67753 = r67751 - r67752;
        return r67753;
}
↓
double f(double x, double n) {
        double r67754 = 1.0;
        double r67755 = n;
        double r67756 = r67754 / r67755;
        double r67757 = -9.914217208649491e-07;
        bool r67758 = r67756 <= r67757;
        double r67759 = 1.639898632791179e-10;
        bool r67760 = r67756 <= r67759;
        double r67761 = !r67760;
        bool r67762 = r67758 || r67761;
        double r67763 = x;
        double r67764 = r67763 + r67754;
        double r67765 = pow(r67764, r67756);
        double r67766 = pow(r67763, r67756);
        double r67767 = r67765 - r67766;
        double r67768 = cbrt(r67767);
        double r67769 = r67768 * r67768;
        double r67770 = r67769 * r67768;
        double r67771 = r67756 / r67763;
        double r67772 = 0.5;
        double r67773 = r67772 / r67755;
        double r67774 = 2.0;
        double r67775 = pow(r67763, r67774);
        double r67776 = r67773 / r67775;
        double r67777 = log(r67763);
        double r67778 = r67777 * r67754;
        double r67779 = pow(r67755, r67774);
        double r67780 = r67763 * r67779;
        double r67781 = r67778 / r67780;
        double r67782 = r67776 - r67781;
        double r67783 = r67771 - r67782;
        double r67784 = r67762 ? r67770 : r67783;
        return r67784;
}

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) < -9.914217208649491e-07 or 1.639898632791179e-10 < (/ 1.0 n)

    1. Initial program 8.4

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

    3. Applied add-cube-cbrt8.4

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

    if -9.914217208649491e-07 < (/ 1.0 n) < 1.639898632791179e-10

    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 32.5

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

    3. Simplified31.9

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

  4. Final simplification21.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -9.91421720864949135 \cdot 10^{-7} \lor \neg \left(\frac{1}{n} \le 1.63989863279117905 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))