Average Error: 29.4 → 19.2
Time: 32.9s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.121470722263304 \cdot 10^{-07}:\\ \;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 4.0245585845241594 \cdot 10^{-07}:\\ \;\;\;\;\left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{1}{x \cdot n}\right) - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -1.121470722263304 \cdot 10^{-07}:\\
\;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\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)\\

\mathbf{elif}\;\frac{1}{n} \le 4.0245585845241594 \cdot 10^{-07}:\\
\;\;\;\;\left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{1}{x \cdot n}\right) - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\\

\mathbf{else}:\\
\;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\

\end{array}
double f(double x, double n) {
        double r3279382 = x;
        double r3279383 = 1.0;
        double r3279384 = r3279382 + r3279383;
        double r3279385 = n;
        double r3279386 = r3279383 / r3279385;
        double r3279387 = pow(r3279384, r3279386);
        double r3279388 = pow(r3279382, r3279386);
        double r3279389 = r3279387 - r3279388;
        return r3279389;
}
↓
double f(double x, double n) {
        double r3279390 = 1.0;
        double r3279391 = n;
        double r3279392 = r3279390 / r3279391;
        double r3279393 = -1.121470722263304e-07;
        bool r3279394 = r3279392 <= r3279393;
        double r3279395 = x;
        double r3279396 = r3279395 + r3279390;
        double r3279397 = pow(r3279396, r3279392);
        double r3279398 = pow(r3279395, r3279392);
        double r3279399 = r3279397 - r3279398;
        double r3279400 = cbrt(r3279399);
        double r3279401 = sqrt(r3279397);
        double r3279402 = sqrt(r3279398);
        double r3279403 = r3279401 - r3279402;
        double r3279404 = r3279402 + r3279401;
        double r3279405 = r3279403 * r3279404;
        double r3279406 = cbrt(r3279405);
        double r3279407 = r3279406 * r3279400;
        double r3279408 = exp(r3279407);
        double r3279409 = log(r3279408);
        double r3279410 = r3279400 * r3279409;
        double r3279411 = 4.0245585845241594e-07;
        bool r3279412 = r3279392 <= r3279411;
        double r3279413 = log(r3279395);
        double r3279414 = r3279395 * r3279391;
        double r3279415 = r3279391 * r3279414;
        double r3279416 = r3279413 / r3279415;
        double r3279417 = r3279390 / r3279414;
        double r3279418 = r3279416 + r3279417;
        double r3279419 = 0.5;
        double r3279420 = r3279419 / r3279391;
        double r3279421 = r3279395 * r3279395;
        double r3279422 = r3279420 / r3279421;
        double r3279423 = r3279418 - r3279422;
        double r3279424 = log1p(r3279395);
        double r3279425 = r3279424 / r3279391;
        double r3279426 = exp(r3279425);
        double r3279427 = r3279426 - r3279398;
        double r3279428 = exp(r3279427);
        double r3279429 = log(r3279428);
        double r3279430 = r3279412 ? r3279423 : r3279429;
        double r3279431 = r3279394 ? r3279410 : r3279430;
        return r3279431;
}

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 3 regimes

  2. if (/ 1 n) < -1.121470722263304e-07

    1. Initial program 0.7

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

    3. Applied add-log-exp0.8

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

    5. Applied add-cube-cbrt0.8

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

    6. Applied exp-prod0.8

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

    7. Applied log-pow0.8

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

    9. Applied add-sqr-sqrt0.8

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

    10. Applied add-sqr-sqrt0.8

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

    11. Applied difference-of-squares0.8

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

    if -1.121470722263304e-07 < (/ 1 n) < 4.0245585845241594e-07

    1. Initial program 45.6

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

    3. Applied add-log-exp45.6

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

    4. Taylor expanded around inf 33.1

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

    5. Simplified33.1

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

    if 4.0245585845241594e-07 < (/ 1 n)

    1. Initial program 24.0

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

    3. Applied add-log-exp24.0

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

    5. Applied add-exp-log24.1

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

    6. Applied pow-exp24.0

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

    7. Simplified1.4

      \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification19.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.121470722263304 \cdot 10^{-07}:\\ \;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 4.0245585845241594 \cdot 10^{-07}:\\ \;\;\;\;\left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{1}{x \cdot n}\right) - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))