Average Error: 28.9 → 21.5
Time: 1.7m
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.099481539779537 \cdot 10^{-08}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 3.9819985237072126 \cdot 10^{-21}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right) - \frac{\frac{\frac{1}{2}}{x \cdot n}}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left({\left(\sqrt{x + 1}\right)}^{\left(\sqrt{\frac{1}{n}}\right)}\right)}^{\left(\sqrt{\frac{1}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -3.099481539779537 \cdot 10^{-08}:\\
\;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\

\mathbf{elif}\;\frac{1}{n} \le 3.9819985237072126 \cdot 10^{-21}:\\
\;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right) - \frac{\frac{\frac{1}{2}}{x \cdot n}}{x}\\

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

\end{array}
double f(double x, double n) {
        double r10026355 = x;
        double r10026356 = 1.0;
        double r10026357 = r10026355 + r10026356;
        double r10026358 = n;
        double r10026359 = r10026356 / r10026358;
        double r10026360 = pow(r10026357, r10026359);
        double r10026361 = pow(r10026355, r10026359);
        double r10026362 = r10026360 - r10026361;
        return r10026362;
}
↓
double f(double x, double n) {
        double r10026363 = 1.0;
        double r10026364 = n;
        double r10026365 = r10026363 / r10026364;
        double r10026366 = -3.099481539779537e-08;
        bool r10026367 = r10026365 <= r10026366;
        double r10026368 = x;
        double r10026369 = r10026368 + r10026363;
        double r10026370 = pow(r10026369, r10026365);
        double r10026371 = pow(r10026368, r10026365);
        double r10026372 = r10026370 - r10026371;
        double r10026373 = 3.9819985237072126e-21;
        bool r10026374 = r10026365 <= r10026373;
        double r10026375 = r10026365 / r10026368;
        double r10026376 = log(r10026368);
        double r10026377 = -r10026376;
        double r10026378 = r10026368 * r10026364;
        double r10026379 = r10026378 * r10026364;
        double r10026380 = r10026377 / r10026379;
        double r10026381 = r10026375 - r10026380;
        double r10026382 = 0.5;
        double r10026383 = r10026382 / r10026378;
        double r10026384 = r10026383 / r10026368;
        double r10026385 = r10026381 - r10026384;
        double r10026386 = sqrt(r10026369);
        double r10026387 = pow(r10026386, r10026365);
        double r10026388 = sqrt(r10026365);
        double r10026389 = pow(r10026386, r10026388);
        double r10026390 = pow(r10026389, r10026388);
        double r10026391 = r10026387 * r10026390;
        double r10026392 = r10026391 - r10026371;
        double r10026393 = r10026374 ? r10026385 : r10026392;
        double r10026394 = r10026367 ? r10026372 : r10026393;
        return r10026394;
}

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) < -3.099481539779537e-08

    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 *-un-lft-identity0.7

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

    4. Applied pow-unpow0.7

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

    5. Simplified0.7

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

    if -3.099481539779537e-08 < (/ 1 n) < 3.9819985237072126e-21

    1. Initial program 44.7

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

    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

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

    5. Applied associate-/r*31.2

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

    if 3.9819985237072126e-21 < (/ 1 n)

    1. Initial program 27.0

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

    3. Applied *-un-lft-identity27.0

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

    4. Applied pow-unpow27.0

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

    5. Simplified27.0

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

    7. Applied add-sqr-sqrt27.0

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

    8. Applied unpow-prod-down27.1

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

    10. Applied add-sqr-sqrt27.1

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

    11. Applied pow-unpow27.1

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

  4. Final simplification21.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.099481539779537 \cdot 10^{-08}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 3.9819985237072126 \cdot 10^{-21}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right) - \frac{\frac{\frac{1}{2}}{x \cdot n}}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left({\left(\sqrt{x + 1}\right)}^{\left(\sqrt{\frac{1}{n}}\right)}\right)}^{\left(\sqrt{\frac{1}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))