Average Error: 29.2 → 21.7
Time: 14.3s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -4.088555932674840565515313109813764352873 \cdot 10^{-10}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.337974556281276381287875419134246685396 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot e^{\log \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -4.088555932674840565515313109813764352873 \cdot 10^{-10}:\\
\;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\

\mathbf{elif}\;\frac{1}{n} \le 2.337974556281276381287875419134246685396 \cdot 10^{-11}:\\
\;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\

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

\end{array}
double f(double x, double n) {
        double r126237 = x;
        double r126238 = 1.0;
        double r126239 = r126237 + r126238;
        double r126240 = n;
        double r126241 = r126238 / r126240;
        double r126242 = pow(r126239, r126241);
        double r126243 = pow(r126237, r126241);
        double r126244 = r126242 - r126243;
        return r126244;
}
↓
double f(double x, double n) {
        double r126245 = 1.0;
        double r126246 = n;
        double r126247 = r126245 / r126246;
        double r126248 = -4.0885559326748406e-10;
        bool r126249 = r126247 <= r126248;
        double r126250 = x;
        double r126251 = r126250 + r126245;
        double r126252 = 2.0;
        double r126253 = r126247 / r126252;
        double r126254 = pow(r126251, r126253);
        double r126255 = pow(r126250, r126253);
        double r126256 = r126254 + r126255;
        double r126257 = r126254 - r126255;
        double r126258 = r126256 * r126257;
        double r126259 = 2.3379745562812764e-11;
        bool r126260 = r126247 <= r126259;
        double r126261 = r126247 / r126250;
        double r126262 = 0.5;
        double r126263 = r126262 / r126246;
        double r126264 = pow(r126250, r126252);
        double r126265 = r126263 / r126264;
        double r126266 = log(r126250);
        double r126267 = r126266 * r126245;
        double r126268 = pow(r126246, r126252);
        double r126269 = r126250 * r126268;
        double r126270 = r126267 / r126269;
        double r126271 = r126265 - r126270;
        double r126272 = r126261 - r126271;
        double r126273 = log(r126257);
        double r126274 = exp(r126273);
        double r126275 = r126256 * r126274;
        double r126276 = r126260 ? r126272 : r126275;
        double r126277 = r126249 ? r126258 : r126276;
        return r126277;
}

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) < -4.0885559326748406e-10

    1. Initial program 1.1

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

    3. Applied sqr-pow1.2

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

    4. Applied sqr-pow1.1

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

    5. Applied difference-of-squares1.1

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

    if -4.0885559326748406e-10 < (/ 1.0 n) < 2.3379745562812764e-11

    1. Initial program 44.8

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

    2. Taylor expanded around inf 32.3

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

    3. Simplified31.6

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

    if 2.3379745562812764e-11 < (/ 1.0 n)

    1. Initial program 25.6

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

    3. Applied sqr-pow25.6

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

    4. Applied sqr-pow25.6

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

    5. Applied difference-of-squares25.6

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

    7. Applied add-exp-log25.6

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

  4. Final simplification21.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -4.088555932674840565515313109813764352873 \cdot 10^{-10}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.337974556281276381287875419134246685396 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot e^{\log \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))