Average Error: 29.5 → 22.1
Time: 16.2s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.01697060178152998838330134390162129420787 \lor \neg \left(\frac{1}{n} \le 1.026143965706023046711342085124602559881 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{1 \cdot \log x}{x \cdot {n}^{2}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -0.01697060178152998838330134390162129420787 \lor \neg \left(\frac{1}{n} \le 1.026143965706023046711342085124602559881 \cdot 10^{-10}\right):\\
\;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{1 \cdot \log x}{x \cdot {n}^{2}}\right)\\

\end{array}
double f(double x, double n) {
        double r92966 = x;
        double r92967 = 1.0;
        double r92968 = r92966 + r92967;
        double r92969 = n;
        double r92970 = r92967 / r92969;
        double r92971 = pow(r92968, r92970);
        double r92972 = pow(r92966, r92970);
        double r92973 = r92971 - r92972;
        return r92973;
}
↓
double f(double x, double n) {
        double r92974 = 1.0;
        double r92975 = n;
        double r92976 = r92974 / r92975;
        double r92977 = -0.01697060178152999;
        bool r92978 = r92976 <= r92977;
        double r92979 = 1.026143965706023e-10;
        bool r92980 = r92976 <= r92979;
        double r92981 = !r92980;
        bool r92982 = r92978 || r92981;
        double r92983 = x;
        double r92984 = r92983 + r92974;
        double r92985 = pow(r92984, r92976);
        double r92986 = cbrt(r92985);
        double r92987 = r92986 * r92986;
        double r92988 = r92987 * r92986;
        double r92989 = pow(r92983, r92976);
        double r92990 = r92988 - r92989;
        double r92991 = r92976 / r92983;
        double r92992 = 0.5;
        double r92993 = r92992 / r92975;
        double r92994 = 2.0;
        double r92995 = pow(r92983, r92994);
        double r92996 = r92993 / r92995;
        double r92997 = log(r92983);
        double r92998 = r92974 * r92997;
        double r92999 = pow(r92975, r92994);
        double r93000 = r92983 * r92999;
        double r93001 = r92998 / r93000;
        double r93002 = r92996 - r93001;
        double r93003 = r92991 - r93002;
        double r93004 = r92982 ? r92990 : r93003;
        return r93004;
}

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 (/ 1.0 n) < -0.01697060178152999 or 1.026143965706023e-10 < (/ 1.0 n)

    1. Initial program 8.6

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

    3. Applied add-cube-cbrt8.6

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

    if -0.01697060178152999 < (/ 1.0 n) < 1.026143965706023e-10

    1. Initial program 44.9

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

    3. Applied add-cube-cbrt44.9

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

    4. Taylor expanded around inf 32.5

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

    5. Simplified32.0

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{1 \cdot \log x}{x \cdot {n}^{2}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification22.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.01697060178152998838330134390162129420787 \lor \neg \left(\frac{1}{n} \le 1.026143965706023046711342085124602559881 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{1 \cdot \log x}{x \cdot {n}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))