Average Error: 29.4 → 21.6
Time: 1.1m
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -767759998.8935319:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right) + \frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)}\\ \mathbf{elif}\;n \le -3.7161554746626 \cdot 10^{-311}:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - e^{\log \left({x}^{\left(\frac{1}{n}\right)}\right)}}\right)\\ \mathbf{elif}\;n \le 8.27878145238031 \cdot 10^{-214}:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1}\right)\\ \mathbf{elif}\;n \le 38062423.250333436:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right) + \frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -767759998.8935319:\\
\;\;\;\;\left(\frac{1}{x \cdot n} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right) + \frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)}\\

\mathbf{elif}\;n \le -3.7161554746626 \cdot 10^{-311}:\\
\;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - e^{\log \left({x}^{\left(\frac{1}{n}\right)}\right)}}\right)\\

\mathbf{elif}\;n \le 8.27878145238031 \cdot 10^{-214}:\\
\;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1}\right)\\

\mathbf{elif}\;n \le 38062423.250333436:\\
\;\;\;\;\left(\sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)}\\

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

\end{array}
double f(double x, double n) {
        double r8526309 = x;
        double r8526310 = 1.0;
        double r8526311 = r8526309 + r8526310;
        double r8526312 = n;
        double r8526313 = r8526310 / r8526312;
        double r8526314 = pow(r8526311, r8526313);
        double r8526315 = pow(r8526309, r8526313);
        double r8526316 = r8526314 - r8526315;
        return r8526316;
}
↓
double f(double x, double n) {
        double r8526317 = n;
        double r8526318 = -767759998.8935319;
        bool r8526319 = r8526317 <= r8526318;
        double r8526320 = 1.0;
        double r8526321 = x;
        double r8526322 = r8526321 * r8526317;
        double r8526323 = r8526320 / r8526322;
        double r8526324 = log(r8526321);
        double r8526325 = -r8526324;
        double r8526326 = r8526322 * r8526317;
        double r8526327 = r8526325 / r8526326;
        double r8526328 = r8526323 - r8526327;
        double r8526329 = -0.5;
        double r8526330 = r8526321 * r8526322;
        double r8526331 = r8526329 / r8526330;
        double r8526332 = r8526328 + r8526331;
        double r8526333 = -3.7161554746626e-311;
        bool r8526334 = r8526317 <= r8526333;
        double r8526335 = r8526320 + r8526321;
        double r8526336 = r8526320 / r8526317;
        double r8526337 = pow(r8526335, r8526336);
        double r8526338 = pow(r8526321, r8526336);
        double r8526339 = log(r8526338);
        double r8526340 = exp(r8526339);
        double r8526341 = r8526337 - r8526340;
        double r8526342 = exp(r8526341);
        double r8526343 = log(r8526342);
        double r8526344 = 8.27878145238031e-214;
        bool r8526345 = r8526317 <= r8526344;
        double r8526346 = r8526337 - r8526320;
        double r8526347 = exp(r8526346);
        double r8526348 = log(r8526347);
        double r8526349 = 38062423.250333436;
        bool r8526350 = r8526317 <= r8526349;
        double r8526351 = r8526337 - r8526338;
        double r8526352 = exp(r8526351);
        double r8526353 = log(r8526352);
        double r8526354 = cbrt(r8526353);
        double r8526355 = r8526354 * r8526354;
        double r8526356 = sqrt(r8526338);
        double r8526357 = r8526356 * r8526356;
        double r8526358 = r8526337 - r8526357;
        double r8526359 = exp(r8526358);
        double r8526360 = log(r8526359);
        double r8526361 = cbrt(r8526360);
        double r8526362 = r8526355 * r8526361;
        double r8526363 = r8526350 ? r8526362 : r8526332;
        double r8526364 = r8526345 ? r8526348 : r8526363;
        double r8526365 = r8526334 ? r8526343 : r8526364;
        double r8526366 = r8526319 ? r8526332 : r8526365;
        return r8526366;
}

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 4 regimes

  2. if n < -767759998.8935319 or 38062423.250333436 < n

    1. Initial program 44.7

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

    3. Applied add-log-exp44.7

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

    4. Taylor expanded around inf 32.7

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

    5. Simplified32.7

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

    if -767759998.8935319 < n < -3.7161554746626e-311

    1. Initial program 1.0

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

    3. Applied add-log-exp1.3

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

    5. Applied add-exp-log1.3

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

    if -3.7161554746626e-311 < n < 8.27878145238031e-214

    1. Initial program 46.2

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

    3. Applied add-log-exp46.2

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

    5. Applied add-exp-log46.2

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

    6. Taylor expanded around 0 17.8

      \[\leadsto \log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\color{blue}{0}}}\right)\]

    if 8.27878145238031e-214 < n < 38062423.250333436

    1. Initial program 17.0

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

    3. Applied add-log-exp17.0

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

    5. Applied add-cube-cbrt17.0

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

    7. Applied add-sqr-sqrt17.0

      \[\leadsto \left(\sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification21.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -767759998.8935319:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right) + \frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)}\\ \mathbf{elif}\;n \le -3.7161554746626 \cdot 10^{-311}:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - e^{\log \left({x}^{\left(\frac{1}{n}\right)}\right)}}\right)\\ \mathbf{elif}\;n \le 8.27878145238031 \cdot 10^{-214}:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1}\right)\\ \mathbf{elif}\;n \le 38062423.250333436:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right) + \frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))