Average Error: 29.6 → 22.1
Time: 28.6s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.9043690733296642 \cdot 10^{-18}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 3.646734676025612 \cdot 10^{-25}:\\ \;\;\;\;\left(\frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)} + \frac{\frac{1}{x}}{n}\right) + \frac{\log x}{n \cdot \left(x \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -2.9043690733296642 \cdot 10^{-18}:\\
\;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\

\mathbf{elif}\;\frac{1}{n} \le 3.646734676025612 \cdot 10^{-25}:\\
\;\;\;\;\left(\frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)} + \frac{\frac{1}{x}}{n}\right) + \frac{\log x}{n \cdot \left(x \cdot n\right)}\\

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

\end{array}
double f(double x, double n) {
        double r1725112 = x;
        double r1725113 = 1.0;
        double r1725114 = r1725112 + r1725113;
        double r1725115 = n;
        double r1725116 = r1725113 / r1725115;
        double r1725117 = pow(r1725114, r1725116);
        double r1725118 = pow(r1725112, r1725116);
        double r1725119 = r1725117 - r1725118;
        return r1725119;
}
↓
double f(double x, double n) {
        double r1725120 = 1.0;
        double r1725121 = n;
        double r1725122 = r1725120 / r1725121;
        double r1725123 = -2.9043690733296642e-18;
        bool r1725124 = r1725122 <= r1725123;
        double r1725125 = x;
        double r1725126 = r1725125 + r1725120;
        double r1725127 = pow(r1725126, r1725122);
        double r1725128 = pow(r1725125, r1725122);
        double r1725129 = r1725127 - r1725128;
        double r1725130 = 3.646734676025612e-25;
        bool r1725131 = r1725122 <= r1725130;
        double r1725132 = -0.5;
        double r1725133 = r1725125 * r1725121;
        double r1725134 = r1725125 * r1725133;
        double r1725135 = r1725132 / r1725134;
        double r1725136 = r1725120 / r1725125;
        double r1725137 = r1725136 / r1725121;
        double r1725138 = r1725135 + r1725137;
        double r1725139 = log(r1725125);
        double r1725140 = r1725121 * r1725133;
        double r1725141 = r1725139 / r1725140;
        double r1725142 = r1725138 + r1725141;
        double r1725143 = r1725131 ? r1725142 : r1725129;
        double r1725144 = r1725124 ? r1725129 : r1725143;
        return r1725144;
}

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 2 regimes

  2. if (/ 1 n) < -2.9043690733296642e-18 or 3.646734676025612e-25 < (/ 1 n)

    1. Initial program 10.8

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

    3. Applied *-un-lft-identity10.8

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

    4. Applied pow-unpow10.8

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

    5. Simplified10.8

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

    if -2.9043690733296642e-18 < (/ 1 n) < 3.646734676025612e-25

    1. Initial program 45.5

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

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

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

    4. Applied pow-unpow45.5

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

    5. Simplified45.5

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

    7. Applied add-log-exp45.5

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

    8. Applied add-log-exp45.5

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

    9. Applied diff-log45.5

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

    10. Simplified45.5

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

    11. Taylor expanded around inf 32.2

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

    12. Simplified31.6

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

  4. Final simplification22.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.9043690733296642 \cdot 10^{-18}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 3.646734676025612 \cdot 10^{-25}:\\ \;\;\;\;\left(\frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)} + \frac{\frac{1}{x}}{n}\right) + \frac{\log x}{n \cdot \left(x \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))