Average Error: 33.0 → 24.4
Time: 21.5s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -7.7456503819782359 \cdot 10^{-30} \lor \neg \left(\frac{1}{n} \le 3.6323271639935607 \cdot 10^{-11}\right):\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, -{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -7.7456503819782359 \cdot 10^{-30} \lor \neg \left(\frac{1}{n} \le 3.6323271639935607 \cdot 10^{-11}\right):\\
\;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, -{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\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 r124297 = x;
        double r124298 = 1.0;
        double r124299 = r124297 + r124298;
        double r124300 = n;
        double r124301 = r124298 / r124300;
        double r124302 = pow(r124299, r124301);
        double r124303 = pow(r124297, r124301);
        double r124304 = r124302 - r124303;
        return r124304;
}
↓
double f(double x, double n) {
        double r124305 = 1.0;
        double r124306 = n;
        double r124307 = r124305 / r124306;
        double r124308 = -7.745650381978236e-30;
        bool r124309 = r124307 <= r124308;
        double r124310 = 3.6323271639935607e-11;
        bool r124311 = r124307 <= r124310;
        double r124312 = !r124311;
        bool r124313 = r124309 || r124312;
        double r124314 = x;
        double r124315 = r124314 + r124305;
        double r124316 = pow(r124315, r124307);
        double r124317 = cbrt(r124316);
        double r124318 = r124317 * r124317;
        double r124319 = cbrt(r124314);
        double r124320 = pow(r124319, r124307);
        double r124321 = r124319 * r124319;
        double r124322 = pow(r124321, r124307);
        double r124323 = r124320 * r124322;
        double r124324 = -r124323;
        double r124325 = fma(r124318, r124317, r124324);
        double r124326 = -r124320;
        double r124327 = r124326 + r124320;
        double r124328 = r124322 * r124327;
        double r124329 = r124325 + r124328;
        double r124330 = cbrt(r124329);
        double r124331 = pow(r124314, r124307);
        double r124332 = r124316 - r124331;
        double r124333 = cbrt(r124332);
        double r124334 = r124330 * r124333;
        double r124335 = r124334 * r124333;
        double r124336 = 1.0;
        double r124337 = r124314 * r124306;
        double r124338 = r124336 / r124337;
        double r124339 = 0.5;
        double r124340 = 2.0;
        double r124341 = pow(r124314, r124340);
        double r124342 = r124341 * r124306;
        double r124343 = r124336 / r124342;
        double r124344 = r124336 / r124314;
        double r124345 = log(r124344);
        double r124346 = pow(r124306, r124340);
        double r124347 = r124314 * r124346;
        double r124348 = r124345 / r124347;
        double r124349 = r124305 * r124348;
        double r124350 = fma(r124339, r124343, r124349);
        double r124351 = -r124350;
        double r124352 = fma(r124305, r124338, r124351);
        double r124353 = r124313 ? r124335 : r124352;
        return r124353;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes

  2. if (/ 1.0 n) < -7.745650381978236e-30 or 3.6323271639935607e-11 < (/ 1.0 n)

    1. Initial program 7.5

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

    3. Applied add-cube-cbrt7.5

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

    5. Applied add-cube-cbrt7.5

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

    6. Applied unpow-prod-down7.5

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

    7. Applied add-cube-cbrt7.5

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

    8. Applied prod-diff7.5

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

    9. Simplified7.5

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

    if -7.745650381978236e-30 < (/ 1.0 n) < 3.6323271639935607e-11

    1. Initial program 44.9

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

    2. Taylor expanded around inf 32.4

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

    3. Simplified32.4

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

  4. Final simplification24.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -7.7456503819782359 \cdot 10^{-30} \lor \neg \left(\frac{1}{n} \le 3.6323271639935607 \cdot 10^{-11}\right):\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, -{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))