Average Error: 29.0 → 22.4
Time: 14.9s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.2682898601339308 \cdot 10^{-17}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.17349982022057744 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{0.5}{{x}^{2} \cdot n} + \frac{-\log x}{x \cdot {n}^{2}} \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} - {x}^{\left(2 \cdot \frac{1}{n}\right)}}{{x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -1.2682898601339308 \cdot 10^{-17}:\\
\;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\

\mathbf{elif}\;\frac{1}{n} \le 1.17349982022057744 \cdot 10^{-10}:\\
\;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{0.5}{{x}^{2} \cdot n} + \frac{-\log x}{x \cdot {n}^{2}} \cdot 1\right)\\

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

\end{array}
double f(double x, double n) {
        double r60282 = x;
        double r60283 = 1.0;
        double r60284 = r60282 + r60283;
        double r60285 = n;
        double r60286 = r60283 / r60285;
        double r60287 = pow(r60284, r60286);
        double r60288 = pow(r60282, r60286);
        double r60289 = r60287 - r60288;
        return r60289;
}
↓
double f(double x, double n) {
        double r60290 = 1.0;
        double r60291 = n;
        double r60292 = r60290 / r60291;
        double r60293 = -1.2682898601339308e-17;
        bool r60294 = r60292 <= r60293;
        double r60295 = x;
        double r60296 = r60295 + r60290;
        double r60297 = pow(r60296, r60292);
        double r60298 = pow(r60295, r60292);
        double r60299 = sqrt(r60298);
        double r60300 = r60299 * r60299;
        double r60301 = r60297 - r60300;
        double r60302 = 1.1734998202205774e-10;
        bool r60303 = r60292 <= r60302;
        double r60304 = r60292 / r60295;
        double r60305 = 0.5;
        double r60306 = 2.0;
        double r60307 = pow(r60295, r60306);
        double r60308 = r60307 * r60291;
        double r60309 = r60305 / r60308;
        double r60310 = log(r60295);
        double r60311 = -r60310;
        double r60312 = pow(r60291, r60306);
        double r60313 = r60295 * r60312;
        double r60314 = r60311 / r60313;
        double r60315 = r60314 * r60290;
        double r60316 = r60309 + r60315;
        double r60317 = r60304 - r60316;
        double r60318 = r60306 * r60292;
        double r60319 = pow(r60296, r60318);
        double r60320 = pow(r60295, r60318);
        double r60321 = r60319 - r60320;
        double r60322 = r60298 + r60297;
        double r60323 = r60321 / r60322;
        double r60324 = r60303 ? r60317 : r60323;
        double r60325 = r60294 ? r60301 : r60324;
        return r60325;
}

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) < -1.2682898601339308e-17

    1. Initial program 2.3

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

    3. Applied add-sqr-sqrt2.4

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

    if -1.2682898601339308e-17 < (/ 1.0 n) < 1.1734998202205774e-10

    1. Initial program 44.4

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

    3. Applied add-log-exp44.4

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

    4. Applied add-log-exp44.4

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

    5. Applied diff-log44.4

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

    6. Simplified44.4

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

    7. Taylor expanded around inf 32.1

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

    8. Simplified31.6

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

    if 1.1734998202205774e-10 < (/ 1.0 n)

    1. Initial program 25.9

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

    3. Applied add-sqr-sqrt25.9

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

    5. Applied flip--29.3

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

    6. Simplified29.2

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

    7. Simplified29.2

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

  4. Final simplification22.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.2682898601339308 \cdot 10^{-17}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.17349982022057744 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{0.5}{{x}^{2} \cdot n} + \frac{-\log x}{x \cdot {n}^{2}} \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} - {x}^{\left(2 \cdot \frac{1}{n}\right)}}{{x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))