Average Error: 29.6 → 22.5
Time: 10.5s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.3388345966815868 \cdot 10^{-10} \lor \neg \left(\frac{1}{n} \le 1.18786278369508663 \cdot 10^{-25}\right):\\ \;\;\;\;\left(\sqrt[3]{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -1.3388345966815868 \cdot 10^{-10} \lor \neg \left(\frac{1}{n} \le 1.18786278369508663 \cdot 10^{-25}\right):\\
\;\;\;\;\left(\sqrt[3]{\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)}} \cdot \sqrt[3]{\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)}}\right) \cdot \sqrt[3]{\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{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\

\end{array}
double f(double x, double n) {
        double r56639 = x;
        double r56640 = 1.0;
        double r56641 = r56639 + r56640;
        double r56642 = n;
        double r56643 = r56640 / r56642;
        double r56644 = pow(r56641, r56643);
        double r56645 = pow(r56639, r56643);
        double r56646 = r56644 - r56645;
        return r56646;
}
↓
double f(double x, double n) {
        double r56647 = 1.0;
        double r56648 = n;
        double r56649 = r56647 / r56648;
        double r56650 = -1.3388345966815868e-10;
        bool r56651 = r56649 <= r56650;
        double r56652 = 1.1878627836950866e-25;
        bool r56653 = r56649 <= r56652;
        double r56654 = !r56653;
        bool r56655 = r56651 || r56654;
        double r56656 = x;
        double r56657 = r56656 + r56647;
        double r56658 = pow(r56657, r56649);
        double r56659 = cbrt(r56658);
        double r56660 = r56659 * r56659;
        double r56661 = r56660 * r56659;
        double r56662 = pow(r56656, r56649);
        double r56663 = r56661 - r56662;
        double r56664 = cbrt(r56663);
        double r56665 = r56664 * r56664;
        double r56666 = r56665 * r56664;
        double r56667 = r56649 / r56656;
        double r56668 = 0.5;
        double r56669 = r56668 / r56648;
        double r56670 = 2.0;
        double r56671 = pow(r56656, r56670);
        double r56672 = r56669 / r56671;
        double r56673 = log(r56656);
        double r56674 = r56673 * r56647;
        double r56675 = pow(r56648, r56670);
        double r56676 = r56656 * r56675;
        double r56677 = r56674 / r56676;
        double r56678 = r56672 - r56677;
        double r56679 = r56667 - r56678;
        double r56680 = r56655 ? r56666 : r56679;
        return r56680;
}

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) < -1.3388345966815868e-10 or 1.1878627836950866e-25 < (/ 1.0 n)

    1. Initial program 9.7

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

    3. Applied add-cube-cbrt9.7

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

    5. Applied add-cube-cbrt9.7

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

    if -1.3388345966815868e-10 < (/ 1.0 n) < 1.1878627836950866e-25

    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 32.8

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

    3. Simplified32.2

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

  4. Final simplification22.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.3388345966815868 \cdot 10^{-10} \lor \neg \left(\frac{1}{n} \le 1.18786278369508663 \cdot 10^{-25}\right):\\ \;\;\;\;\left(\sqrt[3]{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))