Average Error: 29.4 → 21.8
Time: 25.1s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -981129.097746690385974943637847900390625:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} + \frac{1}{\frac{n \cdot \left(x \cdot n\right)}{\log x}}\right) - \frac{\frac{0.5}{n}}{x \cdot x}\\ \mathbf{elif}\;n \le 18355886.857708632946014404296875:\\ \;\;\;\;\log \left(\left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right) \cdot \sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1 \cdot \log x}{{n}^{2} \cdot x} + \frac{\frac{1}{n}}{x}\right) - \log \left(e^{\frac{\frac{0.5}{{x}^{2}}}{n}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -981129.097746690385974943637847900390625:\\
\;\;\;\;\left(\frac{\frac{1}{x}}{n} + \frac{1}{\frac{n \cdot \left(x \cdot n\right)}{\log x}}\right) - \frac{\frac{0.5}{n}}{x \cdot x}\\

\mathbf{elif}\;n \le 18355886.857708632946014404296875:\\
\;\;\;\;\log \left(\left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right) \cdot \sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right) - {x}^{\left(\frac{1}{n}\right)}\\

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

\end{array}
double f(double x, double n) {
        double r53460 = x;
        double r53461 = 1.0;
        double r53462 = r53460 + r53461;
        double r53463 = n;
        double r53464 = r53461 / r53463;
        double r53465 = pow(r53462, r53464);
        double r53466 = pow(r53460, r53464);
        double r53467 = r53465 - r53466;
        return r53467;
}
↓
double f(double x, double n) {
        double r53468 = n;
        double r53469 = -981129.0977466904;
        bool r53470 = r53468 <= r53469;
        double r53471 = 1.0;
        double r53472 = x;
        double r53473 = r53471 / r53472;
        double r53474 = r53473 / r53468;
        double r53475 = r53472 * r53468;
        double r53476 = r53468 * r53475;
        double r53477 = log(r53472);
        double r53478 = r53476 / r53477;
        double r53479 = r53471 / r53478;
        double r53480 = r53474 + r53479;
        double r53481 = 0.5;
        double r53482 = r53481 / r53468;
        double r53483 = r53472 * r53472;
        double r53484 = r53482 / r53483;
        double r53485 = r53480 - r53484;
        double r53486 = 18355886.857708633;
        bool r53487 = r53468 <= r53486;
        double r53488 = r53472 + r53471;
        double r53489 = r53471 / r53468;
        double r53490 = pow(r53488, r53489);
        double r53491 = exp(r53490);
        double r53492 = cbrt(r53491);
        double r53493 = r53492 * r53492;
        double r53494 = r53493 * r53492;
        double r53495 = log(r53494);
        double r53496 = pow(r53472, r53489);
        double r53497 = r53495 - r53496;
        double r53498 = r53471 * r53477;
        double r53499 = 2.0;
        double r53500 = pow(r53468, r53499);
        double r53501 = r53500 * r53472;
        double r53502 = r53498 / r53501;
        double r53503 = r53489 / r53472;
        double r53504 = r53502 + r53503;
        double r53505 = pow(r53472, r53499);
        double r53506 = r53481 / r53505;
        double r53507 = r53506 / r53468;
        double r53508 = exp(r53507);
        double r53509 = log(r53508);
        double r53510 = r53504 - r53509;
        double r53511 = r53487 ? r53497 : r53510;
        double r53512 = r53470 ? r53485 : r53511;
        return r53512;
}

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 n < -981129.0977466904

    1. Initial program 45.2

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

    2. Taylor expanded around inf 32.1

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

    3. Simplified31.4

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

    5. Applied *-un-lft-identity31.4

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

    6. Applied *-un-lft-identity31.4

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

    7. Applied distribute-lft-out31.4

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

    8. Simplified31.4

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

    if -981129.0977466904 < n < 18355886.857708633

    1. Initial program 7.9

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

    3. Applied add-log-exp8.0

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

    4. Simplified8.0

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

    6. Applied add-cube-cbrt8.4

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

    7. Simplified8.4

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

    8. Simplified8.4

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

    if 18355886.857708633 < n

    1. Initial program 45.5

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

    2. Taylor expanded around inf 32.5

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

    3. Simplified31.9

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

    5. Applied add-log-exp32.0

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

    6. Simplified32.0

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

  4. Final simplification21.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -981129.097746690385974943637847900390625:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} + \frac{1}{\frac{n \cdot \left(x \cdot n\right)}{\log x}}\right) - \frac{\frac{0.5}{n}}{x \cdot x}\\ \mathbf{elif}\;n \le 18355886.857708632946014404296875:\\ \;\;\;\;\log \left(\left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right) \cdot \sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1 \cdot \log x}{{n}^{2} \cdot x} + \frac{\frac{1}{n}}{x}\right) - \log \left(e^{\frac{\frac{0.5}{{x}^{2}}}{n}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))