Average Error: 29.3 → 22.2
Time: 1.2m
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -64468894.28699002:\\ \;\;\;\;\sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 3.000508809945158 \cdot 10^{-29}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \left(-\frac{\frac{\log x}{x}}{n \cdot n}\right)\right) - \frac{\frac{1}{2}}{\log \left(e^{n \cdot \left(x \cdot x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -64468894.28699002:\\
\;\;\;\;\sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\\

\mathbf{elif}\;\frac{1}{n} \le 3.000508809945158 \cdot 10^{-29}:\\
\;\;\;\;\left(\frac{\frac{1}{n}}{x} - \left(-\frac{\frac{\log x}{x}}{n \cdot n}\right)\right) - \frac{\frac{1}{2}}{\log \left(e^{n \cdot \left(x \cdot x\right)}\right)}\\

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

\end{array}
double f(double x, double n) {
        double r2315592 = x;
        double r2315593 = 1.0;
        double r2315594 = r2315592 + r2315593;
        double r2315595 = n;
        double r2315596 = r2315593 / r2315595;
        double r2315597 = pow(r2315594, r2315596);
        double r2315598 = pow(r2315592, r2315596);
        double r2315599 = r2315597 - r2315598;
        return r2315599;
}
↓
double f(double x, double n) {
        double r2315600 = 1.0;
        double r2315601 = n;
        double r2315602 = r2315600 / r2315601;
        double r2315603 = -64468894.28699002;
        bool r2315604 = r2315602 <= r2315603;
        double r2315605 = x;
        double r2315606 = r2315605 + r2315600;
        double r2315607 = pow(r2315606, r2315602);
        double r2315608 = pow(r2315605, r2315602);
        double r2315609 = r2315607 - r2315608;
        double r2315610 = r2315609 * r2315609;
        double r2315611 = r2315609 * r2315610;
        double r2315612 = cbrt(r2315611);
        double r2315613 = 3.000508809945158e-29;
        bool r2315614 = r2315602 <= r2315613;
        double r2315615 = r2315602 / r2315605;
        double r2315616 = log(r2315605);
        double r2315617 = r2315616 / r2315605;
        double r2315618 = r2315601 * r2315601;
        double r2315619 = r2315617 / r2315618;
        double r2315620 = -r2315619;
        double r2315621 = r2315615 - r2315620;
        double r2315622 = 0.5;
        double r2315623 = r2315605 * r2315605;
        double r2315624 = r2315601 * r2315623;
        double r2315625 = exp(r2315624);
        double r2315626 = log(r2315625);
        double r2315627 = r2315622 / r2315626;
        double r2315628 = r2315621 - r2315627;
        double r2315629 = r2315614 ? r2315628 : r2315612;
        double r2315630 = r2315604 ? r2315612 : r2315629;
        return r2315630;
}

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 n) < -64468894.28699002 or 3.000508809945158e-29 < (/ 1 n)

    1. Initial program 10.5

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

    3. Applied add-cbrt-cube10.5

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

    if -64468894.28699002 < (/ 1 n) < 3.000508809945158e-29

    1. Initial program 43.5

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

    2. Taylor expanded around inf 31.9

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

    3. Simplified31.9

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

    5. Applied associate-/r*31.3

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

    7. Applied add-log-exp31.1

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

  4. Final simplification22.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -64468894.28699002:\\ \;\;\;\;\sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 3.000508809945158 \cdot 10^{-29}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \left(-\frac{\frac{\log x}{x}}{n \cdot n}\right)\right) - \frac{\frac{1}{2}}{\log \left(e^{n \cdot \left(x \cdot x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))