Average Error: 29.6 → 21.9
Time: 32.6s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.201151584268497640689698496471230740497 \cdot 10^{-12}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left({x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.828861285269525417292226755937066285203 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{1 \cdot \log x}{x \cdot \left(n \cdot n\right)} - \frac{\frac{0.5}{x \cdot x}}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot e^{\log \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}} \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -2.201151584268497640689698496471230740497 \cdot 10^{-12}:\\
\;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left({x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}\\

\mathbf{elif}\;\frac{1}{n} \le 2.828861285269525417292226755937066285203 \cdot 10^{-16}:\\
\;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{1 \cdot \log x}{x \cdot \left(n \cdot n\right)} - \frac{\frac{0.5}{x \cdot x}}{n}\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot e^{\log \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}} \cdot \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)\\

\end{array}
double f(double x, double n) {
        double r2919158 = x;
        double r2919159 = 1.0;
        double r2919160 = r2919158 + r2919159;
        double r2919161 = n;
        double r2919162 = r2919159 / r2919161;
        double r2919163 = pow(r2919160, r2919162);
        double r2919164 = pow(r2919158, r2919162);
        double r2919165 = r2919163 - r2919164;
        return r2919165;
}
↓
double f(double x, double n) {
        double r2919166 = 1.0;
        double r2919167 = n;
        double r2919168 = r2919166 / r2919167;
        double r2919169 = -2.2011515842684976e-12;
        bool r2919170 = r2919168 <= r2919169;
        double r2919171 = x;
        double r2919172 = r2919171 + r2919166;
        double r2919173 = pow(r2919172, r2919168);
        double r2919174 = cbrt(r2919168);
        double r2919175 = r2919174 * r2919174;
        double r2919176 = pow(r2919171, r2919175);
        double r2919177 = pow(r2919176, r2919174);
        double r2919178 = r2919173 - r2919177;
        double r2919179 = 2.8288612852695254e-16;
        bool r2919180 = r2919168 <= r2919179;
        double r2919181 = r2919168 / r2919171;
        double r2919182 = log(r2919171);
        double r2919183 = r2919166 * r2919182;
        double r2919184 = r2919167 * r2919167;
        double r2919185 = r2919171 * r2919184;
        double r2919186 = r2919183 / r2919185;
        double r2919187 = 0.5;
        double r2919188 = r2919171 * r2919171;
        double r2919189 = r2919187 / r2919188;
        double r2919190 = r2919189 / r2919167;
        double r2919191 = r2919186 - r2919190;
        double r2919192 = r2919181 + r2919191;
        double r2919193 = sqrt(r2919173);
        double r2919194 = 2.0;
        double r2919195 = r2919168 / r2919194;
        double r2919196 = pow(r2919171, r2919195);
        double r2919197 = r2919193 + r2919196;
        double r2919198 = r2919193 - r2919196;
        double r2919199 = log(r2919198);
        double r2919200 = exp(r2919199);
        double r2919201 = r2919197 * r2919200;
        double r2919202 = cbrt(r2919201);
        double r2919203 = pow(r2919171, r2919168);
        double r2919204 = r2919173 - r2919203;
        double r2919205 = cbrt(r2919204);
        double r2919206 = r2919205 * r2919205;
        double r2919207 = r2919202 * r2919206;
        double r2919208 = r2919180 ? r2919192 : r2919207;
        double r2919209 = r2919170 ? r2919178 : r2919208;
        return r2919209;
}

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 (/ 1.0 n) < -2.2011515842684976e-12

    1. Initial program 1.5

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

    3. Applied add-cube-cbrt1.5

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

    4. Applied pow-unpow1.5

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

    if -2.2011515842684976e-12 < (/ 1.0 n) < 2.8288612852695254e-16

    1. Initial program 45.2

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

    2. Taylor expanded around inf 32.1

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

    3. Simplified31.5

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

    if 2.8288612852695254e-16 < (/ 1.0 n)

    1. Initial program 27.0

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

    3. Applied add-cube-cbrt27.0

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

    5. Applied sqr-pow27.0

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

    6. Applied add-sqr-sqrt27.0

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

    7. Applied difference-of-squares27.0

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

    9. Applied add-exp-log27.0

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

  4. Final simplification21.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.201151584268497640689698496471230740497 \cdot 10^{-12}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left({x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.828861285269525417292226755937066285203 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{1 \cdot \log x}{x \cdot \left(n \cdot n\right)} - \frac{\frac{0.5}{x \cdot x}}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot e^{\log \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}} \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))