Average Error: 29.7 → 22.1
Time: 33.5s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -13657.800064202142:\\ \;\;\;\;\left(\frac{\frac{-1}{2}}{\left(x \cdot x\right) \cdot n} + \frac{\frac{1}{n}}{x}\right) + \frac{\frac{\log x}{n \cdot n}}{x}\\ \mathbf{elif}\;n \le 1591665190.7901402:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(e^{\frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)}}\right) + \frac{\frac{1}{x}}{n}} \cdot \sqrt{\log \left(e^{\frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)}}\right) + \frac{\frac{1}{x}}{n}} + \frac{\frac{\log x}{n \cdot n}}{x}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -13657.800064202142:\\
\;\;\;\;\left(\frac{\frac{-1}{2}}{\left(x \cdot x\right) \cdot n} + \frac{\frac{1}{n}}{x}\right) + \frac{\frac{\log x}{n \cdot n}}{x}\\

\mathbf{elif}\;n \le 1591665190.7901402:\\
\;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\

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

\end{array}
double f(double x, double n) {
        double r1949143 = x;
        double r1949144 = 1.0;
        double r1949145 = r1949143 + r1949144;
        double r1949146 = n;
        double r1949147 = r1949144 / r1949146;
        double r1949148 = pow(r1949145, r1949147);
        double r1949149 = pow(r1949143, r1949147);
        double r1949150 = r1949148 - r1949149;
        return r1949150;
}
↓
double f(double x, double n) {
        double r1949151 = n;
        double r1949152 = -13657.800064202142;
        bool r1949153 = r1949151 <= r1949152;
        double r1949154 = -0.5;
        double r1949155 = x;
        double r1949156 = r1949155 * r1949155;
        double r1949157 = r1949156 * r1949151;
        double r1949158 = r1949154 / r1949157;
        double r1949159 = 1.0;
        double r1949160 = r1949159 / r1949151;
        double r1949161 = r1949160 / r1949155;
        double r1949162 = r1949158 + r1949161;
        double r1949163 = log(r1949155);
        double r1949164 = r1949151 * r1949151;
        double r1949165 = r1949163 / r1949164;
        double r1949166 = r1949165 / r1949155;
        double r1949167 = r1949162 + r1949166;
        double r1949168 = 1591665190.7901402;
        bool r1949169 = r1949151 <= r1949168;
        double r1949170 = r1949159 + r1949155;
        double r1949171 = pow(r1949170, r1949160);
        double r1949172 = pow(r1949155, r1949160);
        double r1949173 = r1949171 - r1949172;
        double r1949174 = r1949155 * r1949151;
        double r1949175 = r1949155 * r1949174;
        double r1949176 = r1949154 / r1949175;
        double r1949177 = exp(r1949176);
        double r1949178 = log(r1949177);
        double r1949179 = r1949159 / r1949155;
        double r1949180 = r1949179 / r1949151;
        double r1949181 = r1949178 + r1949180;
        double r1949182 = sqrt(r1949181);
        double r1949183 = r1949182 * r1949182;
        double r1949184 = r1949183 + r1949166;
        double r1949185 = r1949169 ? r1949173 : r1949184;
        double r1949186 = r1949153 ? r1949167 : r1949185;
        return r1949186;
}

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 n < -13657.800064202142

    1. Initial program 45.0

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

    3. Applied *-un-lft-identity45.0

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

    4. Applied pow-unpow45.0

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

    5. Simplified45.0

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

    6. Taylor expanded around inf 33.2

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

    7. Simplified32.4

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

    9. Applied add-log-exp32.3

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

    10. Taylor expanded around -inf 33.2

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

    11. Simplified32.4

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

    if -13657.800064202142 < n < 1591665190.7901402

    1. Initial program 8.9

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

    3. Applied *-un-lft-identity8.9

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

    4. Applied pow-unpow8.9

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

    5. Simplified8.9

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

    if 1591665190.7901402 < n

    1. Initial program 45.7

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

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

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

    4. Applied pow-unpow45.7

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

    5. Simplified45.7

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

    6. Taylor expanded around inf 32.3

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

    7. Simplified31.6

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

    9. Applied add-log-exp31.7

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

    11. Applied add-sqr-sqrt31.5

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

  4. Final simplification22.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -13657.800064202142:\\ \;\;\;\;\left(\frac{\frac{-1}{2}}{\left(x \cdot x\right) \cdot n} + \frac{\frac{1}{n}}{x}\right) + \frac{\frac{\log x}{n \cdot n}}{x}\\ \mathbf{elif}\;n \le 1591665190.7901402:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(e^{\frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)}}\right) + \frac{\frac{1}{x}}{n}} \cdot \sqrt{\log \left(e^{\frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)}}\right) + \frac{\frac{1}{x}}{n}} + \frac{\frac{\log x}{n \cdot n}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))