Average Error: 29.3 → 22.2
Time: 8.6s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.62886214871615791 \cdot 10^{-17} \lor \neg \left(\frac{1}{n} \le 4.0466841968524116 \cdot 10^{-15}\right):\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\log \left(e^{\frac{0.5}{{x}^{2} \cdot n}}\right) - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -3.62886214871615791 \cdot 10^{-17} \lor \neg \left(\frac{1}{n} \le 4.0466841968524116 \cdot 10^{-15}\right):\\
\;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\

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

\end{array}
double f(double x, double n) {
        double r38601 = x;
        double r38602 = 1.0;
        double r38603 = r38601 + r38602;
        double r38604 = n;
        double r38605 = r38602 / r38604;
        double r38606 = pow(r38603, r38605);
        double r38607 = pow(r38601, r38605);
        double r38608 = r38606 - r38607;
        return r38608;
}
↓
double f(double x, double n) {
        double r38609 = 1.0;
        double r38610 = n;
        double r38611 = r38609 / r38610;
        double r38612 = -3.628862148716158e-17;
        bool r38613 = r38611 <= r38612;
        double r38614 = 4.046684196852412e-15;
        bool r38615 = r38611 <= r38614;
        double r38616 = !r38615;
        bool r38617 = r38613 || r38616;
        double r38618 = x;
        double r38619 = r38618 + r38609;
        double r38620 = pow(r38619, r38611);
        double r38621 = 3.0;
        double r38622 = pow(r38620, r38621);
        double r38623 = cbrt(r38622);
        double r38624 = pow(r38618, r38611);
        double r38625 = r38623 - r38624;
        double r38626 = r38611 / r38618;
        double r38627 = 0.5;
        double r38628 = 2.0;
        double r38629 = pow(r38618, r38628);
        double r38630 = r38629 * r38610;
        double r38631 = r38627 / r38630;
        double r38632 = exp(r38631);
        double r38633 = log(r38632);
        double r38634 = log(r38618);
        double r38635 = r38634 * r38609;
        double r38636 = pow(r38610, r38628);
        double r38637 = r38618 * r38636;
        double r38638 = r38635 / r38637;
        double r38639 = r38633 - r38638;
        double r38640 = r38626 - r38639;
        double r38641 = r38617 ? r38625 : r38640;
        return r38641;
}

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) < -3.628862148716158e-17 or 4.046684196852412e-15 < (/ 1.0 n)

    1. Initial program 9.5

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

    3. Applied add-cbrt-cube9.6

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

    4. Simplified9.6

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

    if -3.628862148716158e-17 < (/ 1.0 n) < 4.046684196852412e-15

    1. Initial program 44.9

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

    2. Taylor expanded around inf 32.7

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

    3. Simplified32.1

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

    5. Applied add-log-exp32.2

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

    6. Simplified32.2

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

  4. Final simplification22.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.62886214871615791 \cdot 10^{-17} \lor \neg \left(\frac{1}{n} \le 4.0466841968524116 \cdot 10^{-15}\right):\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\log \left(e^{\frac{0.5}{{x}^{2} \cdot n}}\right) - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))