Average Error: 32.1 → 24.6
Time: 18.4s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -10036768523789454 \lor \neg \left(\frac{1}{n} \le 1.86323975548902723 \cdot 10^{-9}\right):\\ \;\;\;\;\sqrt[3]{e^{\sqrt[3]{{\left(\sqrt[3]{{\left(\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)\right)}^{3}}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -10036768523789454 \lor \neg \left(\frac{1}{n} \le 1.86323975548902723 \cdot 10^{-9}\right):\\
\;\;\;\;\sqrt[3]{e^{\sqrt[3]{{\left(\sqrt[3]{{\left(\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)\right)}^{3}}\right)}^{3}}}}\\

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

\end{array}
double f(double x, double n) {
        double r83241 = x;
        double r83242 = 1.0;
        double r83243 = r83241 + r83242;
        double r83244 = n;
        double r83245 = r83242 / r83244;
        double r83246 = pow(r83243, r83245);
        double r83247 = pow(r83241, r83245);
        double r83248 = r83246 - r83247;
        return r83248;
}
↓
double f(double x, double n) {
        double r83249 = 1.0;
        double r83250 = n;
        double r83251 = r83249 / r83250;
        double r83252 = -10036768523789454.0;
        bool r83253 = r83251 <= r83252;
        double r83254 = 1.8632397554890272e-09;
        bool r83255 = r83251 <= r83254;
        double r83256 = !r83255;
        bool r83257 = r83253 || r83256;
        double r83258 = x;
        double r83259 = r83258 + r83249;
        double r83260 = pow(r83259, r83251);
        double r83261 = pow(r83258, r83251);
        double r83262 = r83260 - r83261;
        double r83263 = 3.0;
        double r83264 = pow(r83262, r83263);
        double r83265 = log(r83264);
        double r83266 = pow(r83265, r83263);
        double r83267 = cbrt(r83266);
        double r83268 = pow(r83267, r83263);
        double r83269 = cbrt(r83268);
        double r83270 = exp(r83269);
        double r83271 = cbrt(r83270);
        double r83272 = 1.0;
        double r83273 = r83258 * r83250;
        double r83274 = r83272 / r83273;
        double r83275 = 0.5;
        double r83276 = 2.0;
        double r83277 = pow(r83258, r83276);
        double r83278 = r83277 * r83250;
        double r83279 = r83272 / r83278;
        double r83280 = r83272 / r83258;
        double r83281 = log(r83280);
        double r83282 = pow(r83250, r83276);
        double r83283 = r83258 * r83282;
        double r83284 = r83281 / r83283;
        double r83285 = r83249 * r83284;
        double r83286 = fma(r83275, r83279, r83285);
        double r83287 = -r83286;
        double r83288 = fma(r83249, r83274, r83287);
        double r83289 = r83257 ? r83271 : r83288;
        return r83289;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes

  2. if (/ 1.0 n) < -10036768523789454.0 or 1.8632397554890272e-09 < (/ 1.0 n)

    1. Initial program 2.1

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

    3. Applied add-cbrt-cube2.2

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

    4. Simplified2.2

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

    6. Applied add-exp-log2.2

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

    7. Applied pow-exp2.2

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

    8. Simplified2.2

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

    10. Applied add-cbrt-cube2.2

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

    11. Simplified2.2

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

    13. Applied add-cbrt-cube2.2

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

    14. Simplified2.2

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

    if -10036768523789454.0 < (/ 1.0 n) < 1.8632397554890272e-09

    1. Initial program 44.0

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

    2. Taylor expanded around inf 33.6

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

    3. Simplified33.6

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

  4. Final simplification24.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -10036768523789454 \lor \neg \left(\frac{1}{n} \le 1.86323975548902723 \cdot 10^{-9}\right):\\ \;\;\;\;\sqrt[3]{e^{\sqrt[3]{{\left(\sqrt[3]{{\left(\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)\right)}^{3}}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))