Average Error: 29.0 → 21.9
Time: 17.8s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -80925173752378880 \lor \neg \left(n \le 196016184.31866574\right):\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -80925173752378880 \lor \neg \left(n \le 196016184.31866574\right):\\
\;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\

\mathbf{else}:\\
\;\;\;\;\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)\\

\end{array}
double f(double x, double n) {
        double r59415 = x;
        double r59416 = 1.0;
        double r59417 = r59415 + r59416;
        double r59418 = n;
        double r59419 = r59416 / r59418;
        double r59420 = pow(r59417, r59419);
        double r59421 = pow(r59415, r59419);
        double r59422 = r59420 - r59421;
        return r59422;
}
↓
double f(double x, double n) {
        double r59423 = n;
        double r59424 = -8.092517375237888e+16;
        bool r59425 = r59423 <= r59424;
        double r59426 = 196016184.31866574;
        bool r59427 = r59423 <= r59426;
        double r59428 = !r59427;
        bool r59429 = r59425 || r59428;
        double r59430 = 1.0;
        double r59431 = x;
        double r59432 = r59430 / r59431;
        double r59433 = 1.0;
        double r59434 = r59433 / r59423;
        double r59435 = log(r59431);
        double r59436 = -r59435;
        double r59437 = 2.0;
        double r59438 = pow(r59423, r59437);
        double r59439 = r59436 / r59438;
        double r59440 = r59434 - r59439;
        double r59441 = r59432 * r59440;
        double r59442 = 0.5;
        double r59443 = pow(r59431, r59437);
        double r59444 = r59443 * r59423;
        double r59445 = r59442 / r59444;
        double r59446 = r59441 - r59445;
        double r59447 = r59431 + r59430;
        double r59448 = r59430 / r59423;
        double r59449 = pow(r59447, r59448);
        double r59450 = sqrt(r59449);
        double r59451 = pow(r59431, r59448);
        double r59452 = sqrt(r59451);
        double r59453 = r59450 + r59452;
        double r59454 = r59450 - r59452;
        double r59455 = r59453 * r59454;
        double r59456 = r59429 ? r59446 : r59455;
        return r59456;
}

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 < -8.092517375237888e+16 or 196016184.31866574 < n

    1. Initial program 44.4

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

    2. Taylor expanded around inf 32.1

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

    3. Simplified31.6

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

    if -8.092517375237888e+16 < n < 196016184.31866574

    1. Initial program 9.6

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

    3. Applied add-sqr-sqrt9.7

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

    4. Applied add-sqr-sqrt9.6

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

    5. Applied difference-of-squares9.6

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

  4. Final simplification21.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -80925173752378880 \lor \neg \left(n \le 196016184.31866574\right):\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))