Average Error: 29.6 → 22.5
Time: 11.7s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.02908645454618034170701790230850747320801 \lor \neg \left(\frac{1}{n} \le 1.311945574490123498954663406335030537412 \cdot 10^{-26}\right):\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)}\right) \cdot \sqrt[3]{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)}\right) + 1}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -0.02908645454618034170701790230850747320801 \lor \neg \left(\frac{1}{n} \le 1.311945574490123498954663406335030537412 \cdot 10^{-26}\right):\\
\;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)}\right) \cdot \sqrt[3]{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)}\right) + 1}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)}}\\

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

\end{array}
double f(double x, double n) {
        double r74439 = x;
        double r74440 = 1.0;
        double r74441 = r74439 + r74440;
        double r74442 = n;
        double r74443 = r74440 / r74442;
        double r74444 = pow(r74441, r74443);
        double r74445 = pow(r74439, r74443);
        double r74446 = r74444 - r74445;
        return r74446;
}
↓
double f(double x, double n) {
        double r74447 = 1.0;
        double r74448 = n;
        double r74449 = r74447 / r74448;
        double r74450 = -0.02908645454618034;
        bool r74451 = r74449 <= r74450;
        double r74452 = 1.3119455744901235e-26;
        bool r74453 = r74449 <= r74452;
        double r74454 = !r74453;
        bool r74455 = r74451 || r74454;
        double r74456 = x;
        double r74457 = r74456 + r74447;
        double r74458 = pow(r74457, r74449);
        double r74459 = pow(r74456, r74449);
        double r74460 = log1p(r74459);
        double r74461 = expm1(r74460);
        double r74462 = r74458 - r74461;
        double r74463 = cbrt(r74462);
        double r74464 = r74463 * r74463;
        double r74465 = exp(r74460);
        double r74466 = r74458 - r74465;
        double r74467 = 1.0;
        double r74468 = r74466 + r74467;
        double r74469 = cbrt(r74468);
        double r74470 = r74463 * r74469;
        double r74471 = r74470 * r74463;
        double r74472 = cbrt(r74471);
        double r74473 = r74464 * r74472;
        double r74474 = r74456 * r74448;
        double r74475 = r74467 / r74474;
        double r74476 = 0.5;
        double r74477 = 2.0;
        double r74478 = pow(r74456, r74477);
        double r74479 = r74478 * r74448;
        double r74480 = r74467 / r74479;
        double r74481 = r74467 / r74456;
        double r74482 = log(r74481);
        double r74483 = pow(r74448, r74477);
        double r74484 = r74456 * r74483;
        double r74485 = r74482 / r74484;
        double r74486 = r74447 * r74485;
        double r74487 = fma(r74476, r74480, r74486);
        double r74488 = -r74487;
        double r74489 = fma(r74447, r74475, r74488);
        double r74490 = r74455 ? r74473 : r74489;
        return r74490;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes

  2. if (/ 1.0 n) < -0.02908645454618034 or 1.3119455744901235e-26 < (/ 1.0 n)

    1. Initial program 9.8

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

    3. Applied expm1-log1p-u9.9

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt9.9

      \[\leadsto \color{blue}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)}}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt9.9

      \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)}\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)}}}\]
    8. Using strategy rm

    9. Applied expm1-udef9.9

      \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)}\right) \cdot \sqrt[3]{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)} - 1\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)}}\]

    10. Applied associate--r-9.9

      \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)}\right) \cdot \sqrt[3]{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\color{blue}{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)}\right) + 1}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)}}\]

    if -0.02908645454618034 < (/ 1.0 n) < 1.3119455744901235e-26

    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 32.2

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

    3. Simplified32.2

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

  4. Final simplification22.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.02908645454618034170701790230850747320801 \lor \neg \left(\frac{1}{n} \le 1.311945574490123498954663406335030537412 \cdot 10^{-26}\right):\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)}\right) \cdot \sqrt[3]{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)}\right) + 1}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))