Average Error: 29.5 → 19.5
Time: 35.7s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -4.4434740496799285 \cdot 10^{-09}:\\ \;\;\;\;\sqrt[3]{\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.3833931880144478 \cdot 10^{-21}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\log x}{\left(n \cdot n\right) \cdot x}\right) - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -4.4434740496799285 \cdot 10^{-09}:\\
\;\;\;\;\sqrt[3]{\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right)}}\\

\mathbf{elif}\;\frac{1}{n} \le 1.3833931880144478 \cdot 10^{-21}:\\
\;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\log x}{\left(n \cdot n\right) \cdot x}\right) - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\\

\mathbf{else}:\\
\;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\

\end{array}
double f(double x, double n) {
        double r1984145 = x;
        double r1984146 = 1.0;
        double r1984147 = r1984145 + r1984146;
        double r1984148 = n;
        double r1984149 = r1984146 / r1984148;
        double r1984150 = pow(r1984147, r1984149);
        double r1984151 = pow(r1984145, r1984149);
        double r1984152 = r1984150 - r1984151;
        return r1984152;
}
↓
double f(double x, double n) {
        double r1984153 = 1.0;
        double r1984154 = n;
        double r1984155 = r1984153 / r1984154;
        double r1984156 = -4.4434740496799285e-09;
        bool r1984157 = r1984155 <= r1984156;
        double r1984158 = x;
        double r1984159 = r1984158 + r1984153;
        double r1984160 = pow(r1984159, r1984155);
        double r1984161 = pow(r1984158, r1984155);
        double r1984162 = r1984160 - r1984161;
        double r1984163 = r1984162 * r1984162;
        double r1984164 = exp(r1984162);
        double r1984165 = log(r1984164);
        double r1984166 = r1984162 * r1984165;
        double r1984167 = r1984162 * r1984166;
        double r1984168 = cbrt(r1984167);
        double r1984169 = r1984163 * r1984168;
        double r1984170 = cbrt(r1984169);
        double r1984171 = 1.3833931880144478e-21;
        bool r1984172 = r1984155 <= r1984171;
        double r1984173 = r1984158 * r1984154;
        double r1984174 = r1984153 / r1984173;
        double r1984175 = log(r1984158);
        double r1984176 = r1984154 * r1984154;
        double r1984177 = r1984176 * r1984158;
        double r1984178 = r1984175 / r1984177;
        double r1984179 = r1984174 + r1984178;
        double r1984180 = 0.5;
        double r1984181 = r1984180 / r1984154;
        double r1984182 = r1984158 * r1984158;
        double r1984183 = r1984181 / r1984182;
        double r1984184 = r1984179 - r1984183;
        double r1984185 = log1p(r1984158);
        double r1984186 = r1984185 / r1984154;
        double r1984187 = exp(r1984186);
        double r1984188 = r1984187 - r1984161;
        double r1984189 = r1984172 ? r1984184 : r1984188;
        double r1984190 = r1984157 ? r1984170 : r1984189;
        return r1984190;
}

Error

Bits error versus x

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (/ 1 n) < -4.4434740496799285e-09

    1. Initial program 0.8

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

    3. Applied add-cbrt-cube0.9

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

    5. Applied add-cbrt-cube0.9

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

    7. Applied add-log-exp1.0

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

    if -4.4434740496799285e-09 < (/ 1 n) < 1.3833931880144478e-21

    1. Initial program 45.7

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

    2. Taylor expanded around inf 33.3

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

    3. Simplified33.3

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

    if 1.3833931880144478e-21 < (/ 1 n)

    1. Initial program 27.4

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

    3. Applied add-exp-log27.4

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

    4. Applied pow-exp27.4

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

    5. Simplified5.8

      \[\leadsto e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification19.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -4.4434740496799285 \cdot 10^{-09}:\\ \;\;\;\;\sqrt[3]{\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.3833931880144478 \cdot 10^{-21}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\log x}{\left(n \cdot n\right) \cdot x}\right) - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))