Average Error: 29.4 → 19.2
Time: 2.9m
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.6458580981343804 \cdot 10^{-16}:\\ \;\;\;\;\log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left({x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left({x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 6.469269257276286 \cdot 10^{-11}:\\ \;\;\;\;(\left(\frac{\frac{-1}{2}}{x}\right) \cdot \left(\frac{1}{x \cdot n}\right) + \left(\frac{1}{x \cdot n} - \left(-\frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\right))_*\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -1.6458580981343804 \cdot 10^{-16}:\\
\;\;\;\;\log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left({x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left({x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}}}\right)\\

\mathbf{elif}\;\frac{1}{n} \le 6.469269257276286 \cdot 10^{-11}:\\
\;\;\;\;(\left(\frac{\frac{-1}{2}}{x}\right) \cdot \left(\frac{1}{x \cdot n}\right) + \left(\frac{1}{x \cdot n} - \left(-\frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\right))_*\\

\mathbf{else}:\\
\;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\

\end{array}
double f(double x, double n) {
        double r14489526 = x;
        double r14489527 = 1.0;
        double r14489528 = r14489526 + r14489527;
        double r14489529 = n;
        double r14489530 = r14489527 / r14489529;
        double r14489531 = pow(r14489528, r14489530);
        double r14489532 = pow(r14489526, r14489530);
        double r14489533 = r14489531 - r14489532;
        return r14489533;
}
↓
double f(double x, double n) {
        double r14489534 = 1.0;
        double r14489535 = n;
        double r14489536 = r14489534 / r14489535;
        double r14489537 = -1.6458580981343804e-16;
        bool r14489538 = r14489536 <= r14489537;
        double r14489539 = x;
        double r14489540 = r14489539 + r14489534;
        double r14489541 = 2.0;
        double r14489542 = r14489536 / r14489541;
        double r14489543 = pow(r14489540, r14489542);
        double r14489544 = r14489543 * r14489543;
        double r14489545 = cbrt(r14489536);
        double r14489546 = r14489545 * r14489545;
        double r14489547 = pow(r14489539, r14489546);
        double r14489548 = pow(r14489547, r14489545);
        double r14489549 = r14489544 - r14489548;
        double r14489550 = exp(r14489549);
        double r14489551 = sqrt(r14489550);
        double r14489552 = log(r14489551);
        double r14489553 = r14489552 + r14489552;
        double r14489554 = 6.469269257276286e-11;
        bool r14489555 = r14489536 <= r14489554;
        double r14489556 = -0.5;
        double r14489557 = r14489556 / r14489539;
        double r14489558 = r14489539 * r14489535;
        double r14489559 = r14489534 / r14489558;
        double r14489560 = log(r14489539);
        double r14489561 = r14489535 * r14489558;
        double r14489562 = r14489560 / r14489561;
        double r14489563 = -r14489562;
        double r14489564 = r14489559 - r14489563;
        double r14489565 = fma(r14489557, r14489559, r14489564);
        double r14489566 = log1p(r14489539);
        double r14489567 = r14489566 / r14489535;
        double r14489568 = exp(r14489567);
        double r14489569 = pow(r14489539, r14489536);
        double r14489570 = r14489568 - r14489569;
        double r14489571 = r14489555 ? r14489565 : r14489570;
        double r14489572 = r14489538 ? r14489553 : r14489571;
        return r14489572;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (/ 1 n) < -1.6458580981343804e-16

    1. Initial program 2.3

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

    3. Applied add-cube-cbrt2.3

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

    4. Applied pow-unpow2.4

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

    6. Applied sqr-pow2.5

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}} - {\left({x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}\]
    7. Using strategy rm

    8. Applied add-log-exp2.7

      \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left({x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}}\right)}\]
    9. Using strategy rm

    10. Applied add-sqr-sqrt2.7

      \[\leadsto \log \color{blue}{\left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left({x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}}} \cdot \sqrt{e^{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left({x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}}}\right)}\]

    11. Applied log-prod2.7

      \[\leadsto \color{blue}{\log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left({x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left({x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}}}\right)}\]

    if -1.6458580981343804e-16 < (/ 1 n) < 6.469269257276286e-11

    1. Initial program 45.1

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

    2. Taylor expanded around inf 31.8

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

    3. Simplified31.8

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

    if 6.469269257276286e-11 < (/ 1 n)

    1. Initial program 23.5

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

    3. Applied add-exp-log23.5

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

    4. Applied pow-exp23.5

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

    5. Simplified2.8

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

  4. Final simplification19.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.6458580981343804 \cdot 10^{-16}:\\ \;\;\;\;\log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left({x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left({x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 6.469269257276286 \cdot 10^{-11}:\\ \;\;\;\;(\left(\frac{\frac{-1}{2}}{x}\right) \cdot \left(\frac{1}{x \cdot n}\right) + \left(\frac{1}{x \cdot n} - \left(-\frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\right))_*\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))