Average Error: 29.4 → 21.8
Time: 10.9s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.6635811141354433 \cdot 10^{-4} \lor \neg \left(\frac{1}{n} \le 6.22124734048439072 \cdot 10^{-16}\right):\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \mathsf{fma}\left(-0.25, \frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}, \frac{\frac{0.5}{n}}{x}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -6.6635811141354433 \cdot 10^{-4} \lor \neg \left(\frac{1}{n} \le 6.22124734048439072 \cdot 10^{-16}\right):\\
\;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \mathsf{fma}\left(-0.25, \frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}, \frac{\frac{0.5}{n}}{x}\right)\\

\end{array}
double f(double x, double n) {
        double r61228 = x;
        double r61229 = 1.0;
        double r61230 = r61228 + r61229;
        double r61231 = n;
        double r61232 = r61229 / r61231;
        double r61233 = pow(r61230, r61232);
        double r61234 = pow(r61228, r61232);
        double r61235 = r61233 - r61234;
        return r61235;
}
↓
double f(double x, double n) {
        double r61236 = 1.0;
        double r61237 = n;
        double r61238 = r61236 / r61237;
        double r61239 = -0.0006663581114135443;
        bool r61240 = r61238 <= r61239;
        double r61241 = 6.221247340484391e-16;
        bool r61242 = r61238 <= r61241;
        double r61243 = !r61242;
        bool r61244 = r61240 || r61243;
        double r61245 = x;
        double r61246 = r61245 + r61236;
        double r61247 = 2.0;
        double r61248 = r61238 / r61247;
        double r61249 = pow(r61246, r61248);
        double r61250 = pow(r61245, r61248);
        double r61251 = r61249 + r61250;
        double r61252 = r61249 - r61250;
        double r61253 = 3.0;
        double r61254 = pow(r61252, r61253);
        double r61255 = cbrt(r61254);
        double r61256 = r61251 * r61255;
        double r61257 = 0.25;
        double r61258 = -r61257;
        double r61259 = 1.0;
        double r61260 = pow(r61245, r61247);
        double r61261 = r61260 * r61237;
        double r61262 = r61259 / r61261;
        double r61263 = r61259 / r61245;
        double r61264 = log(r61263);
        double r61265 = pow(r61237, r61247);
        double r61266 = r61245 * r61265;
        double r61267 = r61264 / r61266;
        double r61268 = r61262 + r61267;
        double r61269 = 0.5;
        double r61270 = r61269 / r61237;
        double r61271 = r61270 / r61245;
        double r61272 = fma(r61258, r61268, r61271);
        double r61273 = r61251 * r61272;
        double r61274 = r61244 ? r61256 : r61273;
        return r61274;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes

  2. if (/ 1.0 n) < -0.0006663581114135443 or 6.221247340484391e-16 < (/ 1.0 n)

    1. Initial program 8.3

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

    3. Applied sqr-pow8.3

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

    4. Applied sqr-pow8.3

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

    5. Applied difference-of-squares8.3

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

    7. Applied add-cbrt-cube8.3

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\sqrt[3]{\left(\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}}\]

    8. Simplified8.3

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \sqrt[3]{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}}}\]

    if -0.0006663581114135443 < (/ 1.0 n) < 6.221247340484391e-16

    1. Initial program 45.4

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

    3. Applied sqr-pow45.5

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

    4. Applied sqr-pow45.4

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

    5. Applied difference-of-squares45.4

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

    6. Taylor expanded around inf 32.7

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\left(0.5 \cdot \frac{1}{x \cdot n} - \left(0.25 \cdot \frac{1}{{x}^{2} \cdot n} + 0.25 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)}\]

    7. Simplified32.1

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.25, \frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}, \frac{\frac{0.5}{n}}{x}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification21.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.6635811141354433 \cdot 10^{-4} \lor \neg \left(\frac{1}{n} \le 6.22124734048439072 \cdot 10^{-16}\right):\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \mathsf{fma}\left(-0.25, \frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}, \frac{\frac{0.5}{n}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))