Average Error: 29.8 → 19.3
Time: 3.1m
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -62.577189808211486:\\ \;\;\;\;(\left(\frac{\log x}{n}\right) \cdot \left(\frac{\frac{1}{n}}{x}\right) + \left((\left(\frac{\frac{\frac{1}{n}}{x}}{x}\right) \cdot \frac{-1}{2} + \left(\frac{\frac{1}{n}}{x}\right))_*\right))_*\\ \mathbf{elif}\;n \le -5.6166853937121 \cdot 10^{-310}:\\ \;\;\;\;{e}^{\left(\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\\ \mathbf{elif}\;n \le 5977604505.843376:\\ \;\;\;\;{e}^{\left(\log \left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\log x}{n}\right) \cdot \left(\frac{\frac{1}{n}}{x}\right) + \left((\left(\frac{\frac{\frac{1}{n}}{x}}{x}\right) \cdot \frac{-1}{2} + \left(\frac{\frac{1}{n}}{x}\right))_*\right))_*\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -62.577189808211486:\\
\;\;\;\;(\left(\frac{\log x}{n}\right) \cdot \left(\frac{\frac{1}{n}}{x}\right) + \left((\left(\frac{\frac{\frac{1}{n}}{x}}{x}\right) \cdot \frac{-1}{2} + \left(\frac{\frac{1}{n}}{x}\right))_*\right))_*\\

\mathbf{elif}\;n \le -5.6166853937121 \cdot 10^{-310}:\\
\;\;\;\;{e}^{\left(\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\\

\mathbf{elif}\;n \le 5977604505.843376:\\
\;\;\;\;{e}^{\left(\log \left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;(\left(\frac{\log x}{n}\right) \cdot \left(\frac{\frac{1}{n}}{x}\right) + \left((\left(\frac{\frac{\frac{1}{n}}{x}}{x}\right) \cdot \frac{-1}{2} + \left(\frac{\frac{1}{n}}{x}\right))_*\right))_*\\

\end{array}
double f(double x, double n) {
        double r13630186 = x;
        double r13630187 = 1.0;
        double r13630188 = r13630186 + r13630187;
        double r13630189 = n;
        double r13630190 = r13630187 / r13630189;
        double r13630191 = pow(r13630188, r13630190);
        double r13630192 = pow(r13630186, r13630190);
        double r13630193 = r13630191 - r13630192;
        return r13630193;
}
↓
double f(double x, double n) {
        double r13630194 = n;
        double r13630195 = -62.577189808211486;
        bool r13630196 = r13630194 <= r13630195;
        double r13630197 = x;
        double r13630198 = log(r13630197);
        double r13630199 = r13630198 / r13630194;
        double r13630200 = 1.0;
        double r13630201 = r13630200 / r13630194;
        double r13630202 = r13630201 / r13630197;
        double r13630203 = r13630202 / r13630197;
        double r13630204 = -0.5;
        double r13630205 = fma(r13630203, r13630204, r13630202);
        double r13630206 = fma(r13630199, r13630202, r13630205);
        double r13630207 = -5.6166853937121e-310;
        bool r13630208 = r13630194 <= r13630207;
        double r13630209 = exp(1.0);
        double r13630210 = r13630197 + r13630200;
        double r13630211 = pow(r13630210, r13630201);
        double r13630212 = pow(r13630197, r13630201);
        double r13630213 = r13630211 - r13630212;
        double r13630214 = log(r13630213);
        double r13630215 = pow(r13630209, r13630214);
        double r13630216 = 5977604505.843376;
        bool r13630217 = r13630194 <= r13630216;
        double r13630218 = log1p(r13630197);
        double r13630219 = r13630218 / r13630194;
        double r13630220 = exp(r13630219);
        double r13630221 = r13630220 - r13630212;
        double r13630222 = log(r13630221);
        double r13630223 = pow(r13630209, r13630222);
        double r13630224 = r13630217 ? r13630223 : r13630206;
        double r13630225 = r13630208 ? r13630215 : r13630224;
        double r13630226 = r13630196 ? r13630206 : r13630225;
        return r13630226;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if n < -62.577189808211486 or 5977604505.843376 < n

    1. Initial program 45.1

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

    3. Applied add-exp-log45.7

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

    5. Applied *-un-lft-identity45.7

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

    6. Applied exp-prod45.7

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

    7. Simplified45.7

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

    8. Taylor expanded around inf 48.5

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

    9. Simplified32.5

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

    if -62.577189808211486 < n < -5.6166853937121e-310

    1. Initial program 0.2

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

    3. Applied add-exp-log0.4

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

    5. Applied *-un-lft-identity0.4

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

    6. Applied exp-prod0.4

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

    7. Simplified0.4

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

    if -5.6166853937121e-310 < n < 5977604505.843376

    1. Initial program 25.7

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

    3. Applied add-exp-log25.7

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

    5. Applied *-un-lft-identity25.7

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

    6. Applied exp-prod25.7

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

    7. Simplified25.7

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

    9. Applied add-exp-log25.7

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

    10. Applied pow-exp25.7

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

    11. Simplified2.2

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

  4. Final simplification19.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -62.577189808211486:\\ \;\;\;\;(\left(\frac{\log x}{n}\right) \cdot \left(\frac{\frac{1}{n}}{x}\right) + \left((\left(\frac{\frac{\frac{1}{n}}{x}}{x}\right) \cdot \frac{-1}{2} + \left(\frac{\frac{1}{n}}{x}\right))_*\right))_*\\ \mathbf{elif}\;n \le -5.6166853937121 \cdot 10^{-310}:\\ \;\;\;\;{e}^{\left(\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\\ \mathbf{elif}\;n \le 5977604505.843376:\\ \;\;\;\;{e}^{\left(\log \left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\log x}{n}\right) \cdot \left(\frac{\frac{1}{n}}{x}\right) + \left((\left(\frac{\frac{\frac{1}{n}}{x}}{x}\right) \cdot \frac{-1}{2} + \left(\frac{\frac{1}{n}}{x}\right))_*\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))