Average Error: 28.8 → 22.3
Time: 30.1s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -54631167.2909455597400665283203125 \lor \neg \left(n \le 1197072613876769056513589248\right):\\ \;\;\;\;\frac{1}{x \cdot n} - \mathsf{fma}\left(\frac{-\log x}{x \cdot {n}^{2}}, 1, \frac{0.5}{{x}^{2} \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -54631167.2909455597400665283203125 \lor \neg \left(n \le 1197072613876769056513589248\right):\\
\;\;\;\;\frac{1}{x \cdot n} - \mathsf{fma}\left(\frac{-\log x}{x \cdot {n}^{2}}, 1, \frac{0.5}{{x}^{2} \cdot n}\right)\\

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

\end{array}
double f(double x, double n) {
        double r68436 = x;
        double r68437 = 1.0;
        double r68438 = r68436 + r68437;
        double r68439 = n;
        double r68440 = r68437 / r68439;
        double r68441 = pow(r68438, r68440);
        double r68442 = pow(r68436, r68440);
        double r68443 = r68441 - r68442;
        return r68443;
}
↓
double f(double x, double n) {
        double r68444 = n;
        double r68445 = -54631167.29094556;
        bool r68446 = r68444 <= r68445;
        double r68447 = 1.197072613876769e+27;
        bool r68448 = r68444 <= r68447;
        double r68449 = !r68448;
        bool r68450 = r68446 || r68449;
        double r68451 = 1.0;
        double r68452 = x;
        double r68453 = r68452 * r68444;
        double r68454 = r68451 / r68453;
        double r68455 = log(r68452);
        double r68456 = -r68455;
        double r68457 = 2.0;
        double r68458 = pow(r68444, r68457);
        double r68459 = r68452 * r68458;
        double r68460 = r68456 / r68459;
        double r68461 = 0.5;
        double r68462 = pow(r68452, r68457);
        double r68463 = r68462 * r68444;
        double r68464 = r68461 / r68463;
        double r68465 = fma(r68460, r68451, r68464);
        double r68466 = r68454 - r68465;
        double r68467 = r68452 + r68451;
        double r68468 = r68451 / r68444;
        double r68469 = pow(r68467, r68468);
        double r68470 = pow(r68452, r68468);
        double r68471 = cbrt(r68470);
        double r68472 = r68471 * r68471;
        double r68473 = r68472 * r68471;
        double r68474 = r68469 - r68473;
        double r68475 = exp(r68474);
        double r68476 = log(r68475);
        double r68477 = r68450 ? r68466 : r68476;
        return r68477;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes

  2. if n < -54631167.29094556 or 1.197072613876769e+27 < n

    1. Initial program 44.0

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

    3. Applied add-log-exp44.0

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

    4. Applied add-log-exp44.0

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

    5. Applied diff-log44.0

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

    6. Simplified44.0

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

    8. Applied add-cube-cbrt44.0

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

    9. Taylor expanded around inf 32.1

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

    10. Simplified32.1

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

    if -54631167.29094556 < n < 1.197072613876769e+27

    1. Initial program 9.9

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

    3. Applied add-log-exp10.0

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

    4. Applied add-log-exp10.0

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

    5. Applied diff-log10.0

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

    6. Simplified10.0

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

    8. Applied add-cube-cbrt10.1

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

  4. Final simplification22.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -54631167.2909455597400665283203125 \lor \neg \left(n \le 1197072613876769056513589248\right):\\ \;\;\;\;\frac{1}{x \cdot n} - \mathsf{fma}\left(\frac{-\log x}{x \cdot {n}^{2}}, 1, \frac{0.5}{{x}^{2} \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))