Average Error: 29.2 → 21.7
Time: 27.6s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -1313.139082756415291441953741014003753662 \lor \neg \left(n \le 56897569.691876567900180816650390625\right):\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{1 \cdot \left(-\log x\right)}{x \cdot {n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{6}} \cdot \sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\right)}^{3}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -1313.139082756415291441953741014003753662 \lor \neg \left(n \le 56897569.691876567900180816650390625\right):\\
\;\;\;\;\left(\frac{1}{x \cdot n} - \frac{1 \cdot \left(-\log x\right)}{x \cdot {n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{6}} \cdot \sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\right)}^{3}}\\

\end{array}
double f(double x, double n) {
        double r74470 = x;
        double r74471 = 1.0;
        double r74472 = r74470 + r74471;
        double r74473 = n;
        double r74474 = r74471 / r74473;
        double r74475 = pow(r74472, r74474);
        double r74476 = pow(r74470, r74474);
        double r74477 = r74475 - r74476;
        return r74477;
}
↓
double f(double x, double n) {
        double r74478 = n;
        double r74479 = -1313.1390827564153;
        bool r74480 = r74478 <= r74479;
        double r74481 = 56897569.69187657;
        bool r74482 = r74478 <= r74481;
        double r74483 = !r74482;
        bool r74484 = r74480 || r74483;
        double r74485 = 1.0;
        double r74486 = x;
        double r74487 = r74486 * r74478;
        double r74488 = r74485 / r74487;
        double r74489 = log(r74486);
        double r74490 = -r74489;
        double r74491 = r74485 * r74490;
        double r74492 = 2.0;
        double r74493 = pow(r74478, r74492);
        double r74494 = r74486 * r74493;
        double r74495 = r74491 / r74494;
        double r74496 = r74488 - r74495;
        double r74497 = 0.5;
        double r74498 = pow(r74486, r74492);
        double r74499 = r74498 * r74478;
        double r74500 = r74497 / r74499;
        double r74501 = r74496 - r74500;
        double r74502 = r74486 + r74485;
        double r74503 = r74485 / r74478;
        double r74504 = pow(r74502, r74503);
        double r74505 = pow(r74486, r74503);
        double r74506 = r74504 - r74505;
        double r74507 = cbrt(r74506);
        double r74508 = 6.0;
        double r74509 = pow(r74507, r74508);
        double r74510 = cbrt(r74509);
        double r74511 = r74503 / r74492;
        double r74512 = pow(r74502, r74511);
        double r74513 = sqrt(r74505);
        double r74514 = r74512 + r74513;
        double r74515 = r74512 - r74513;
        double r74516 = r74514 * r74515;
        double r74517 = cbrt(r74516);
        double r74518 = r74510 * r74517;
        double r74519 = 3.0;
        double r74520 = pow(r74518, r74519);
        double r74521 = cbrt(r74520);
        double r74522 = r74484 ? r74501 : r74521;
        return r74522;
}

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 n < -1313.1390827564153 or 56897569.69187657 < n

    1. Initial program 44.8

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

    2. Taylor expanded around inf 31.8

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

    3. Simplified31.8

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

    if -1313.1390827564153 < n < 56897569.69187657

    1. Initial program 7.7

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

    3. Applied add-cbrt-cube7.8

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

    4. Simplified7.8

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

    6. Applied add-cube-cbrt7.8

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

    7. Simplified7.8

      \[\leadsto \sqrt[3]{{\left(\color{blue}{\sqrt[3]{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{6}}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt7.8

      \[\leadsto \sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{6}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}}\right)}^{3}}\]

    10. Applied sqr-pow7.8

      \[\leadsto \sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{6}} \cdot \sqrt[3]{\color{blue}{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)}^{3}}\]

    11. Applied difference-of-squares7.8

      \[\leadsto \sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{6}} \cdot \sqrt[3]{\color{blue}{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}}\right)}^{3}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification21.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1313.139082756415291441953741014003753662 \lor \neg \left(n \le 56897569.691876567900180816650390625\right):\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{1 \cdot \left(-\log x\right)}{x \cdot {n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{6}} \cdot \sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))