Average Error: 29.2 → 22.4
Time: 30.0s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.9414380402982056 \cdot 10^{-06}:\\ \;\;\;\;\log \left(\frac{e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 9.047336182507362 \cdot 10^{-24}:\\ \;\;\;\;\left(\frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)} + \frac{\log x}{\left(x \cdot n\right) \cdot n}\right) + \frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \sqrt[3]{\log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -3.9414380402982056 \cdot 10^{-06}:\\
\;\;\;\;\log \left(\frac{e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)\\

\mathbf{elif}\;\frac{1}{n} \le 9.047336182507362 \cdot 10^{-24}:\\
\;\;\;\;\left(\frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)} + \frac{\log x}{\left(x \cdot n\right) \cdot n}\right) + \frac{\frac{1}{n}}{x}\\

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

\end{array}
double f(double x, double n) {
        double r2348090 = x;
        double r2348091 = 1.0;
        double r2348092 = r2348090 + r2348091;
        double r2348093 = n;
        double r2348094 = r2348091 / r2348093;
        double r2348095 = pow(r2348092, r2348094);
        double r2348096 = pow(r2348090, r2348094);
        double r2348097 = r2348095 - r2348096;
        return r2348097;
}
↓
double f(double x, double n) {
        double r2348098 = 1.0;
        double r2348099 = n;
        double r2348100 = r2348098 / r2348099;
        double r2348101 = -3.9414380402982056e-06;
        bool r2348102 = r2348100 <= r2348101;
        double r2348103 = x;
        double r2348104 = r2348103 + r2348098;
        double r2348105 = pow(r2348104, r2348100);
        double r2348106 = r2348105 * r2348105;
        double r2348107 = r2348105 * r2348106;
        double r2348108 = cbrt(r2348107);
        double r2348109 = exp(r2348108);
        double r2348110 = pow(r2348103, r2348100);
        double r2348111 = exp(r2348110);
        double r2348112 = r2348109 / r2348111;
        double r2348113 = log(r2348112);
        double r2348114 = 9.047336182507362e-24;
        bool r2348115 = r2348100 <= r2348114;
        double r2348116 = -0.5;
        double r2348117 = r2348103 * r2348099;
        double r2348118 = r2348103 * r2348117;
        double r2348119 = r2348116 / r2348118;
        double r2348120 = log(r2348103);
        double r2348121 = r2348117 * r2348099;
        double r2348122 = r2348120 / r2348121;
        double r2348123 = r2348119 + r2348122;
        double r2348124 = r2348100 / r2348103;
        double r2348125 = r2348123 + r2348124;
        double r2348126 = r2348108 - r2348110;
        double r2348127 = cbrt(r2348126);
        double r2348128 = r2348127 * r2348127;
        double r2348129 = exp(r2348128);
        double r2348130 = log(r2348129);
        double r2348131 = exp(r2348126);
        double r2348132 = log(r2348131);
        double r2348133 = cbrt(r2348132);
        double r2348134 = r2348130 * r2348133;
        double r2348135 = r2348115 ? r2348125 : r2348134;
        double r2348136 = r2348102 ? r2348113 : r2348135;
        return r2348136;
}

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 n) < -3.9414380402982056e-06

    1. Initial program 0.5

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

    3. Applied add-cbrt-cube0.6

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

    5. Applied add-log-exp0.8

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

    6. Applied add-log-exp0.7

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

    7. Applied diff-log0.7

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

    if -3.9414380402982056e-06 < (/ 1 n) < 9.047336182507362e-24

    1. Initial program 44.2

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

    3. Applied add-cbrt-cube44.3

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

    5. Applied add-log-exp44.3

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

    6. Taylor expanded around inf 32.5

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

    7. Simplified32.0

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

    if 9.047336182507362e-24 < (/ 1 n)

    1. Initial program 28.7

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

    3. Applied add-cbrt-cube28.7

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

    5. Applied add-log-exp28.8

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

    7. Applied add-cube-cbrt28.8

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

    8. Applied exp-prod28.8

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

    9. Applied log-pow28.7

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

    11. Applied add-log-exp28.7

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

  4. Final simplification22.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.9414380402982056 \cdot 10^{-06}:\\ \;\;\;\;\log \left(\frac{e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 9.047336182507362 \cdot 10^{-24}:\\ \;\;\;\;\left(\frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)} + \frac{\log x}{\left(x \cdot n\right) \cdot n}\right) + \frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \sqrt[3]{\log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))