Average Error: 33.2 → 23.5
Time: 12.9s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.4934594501765746 \cdot 10^{-7}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.780710866682253 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{1 \cdot \log x}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} - {x}^{\left(2 \cdot \frac{1}{n}\right)}}{{x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -2.4934594501765746 \cdot 10^{-7}:\\
\;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\

\mathbf{elif}\;\frac{1}{n} \le 1.780710866682253 \cdot 10^{-5}:\\
\;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{1 \cdot \log x}{x \cdot {n}^{2}}\right)\\

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

\end{array}
double f(double x, double n) {
        double r58095 = x;
        double r58096 = 1.0;
        double r58097 = r58095 + r58096;
        double r58098 = n;
        double r58099 = r58096 / r58098;
        double r58100 = pow(r58097, r58099);
        double r58101 = pow(r58095, r58099);
        double r58102 = r58100 - r58101;
        return r58102;
}
↓
double f(double x, double n) {
        double r58103 = 1.0;
        double r58104 = n;
        double r58105 = r58103 / r58104;
        double r58106 = -2.4934594501765746e-07;
        bool r58107 = r58105 <= r58106;
        double r58108 = x;
        double r58109 = r58108 + r58103;
        double r58110 = pow(r58109, r58105);
        double r58111 = pow(r58108, r58105);
        double r58112 = r58110 - r58111;
        double r58113 = exp(r58112);
        double r58114 = log(r58113);
        double r58115 = 1.780710866682253e-05;
        bool r58116 = r58105 <= r58115;
        double r58117 = r58105 / r58108;
        double r58118 = 0.5;
        double r58119 = r58118 / r58104;
        double r58120 = 2.0;
        double r58121 = pow(r58108, r58120);
        double r58122 = r58119 / r58121;
        double r58123 = log(r58108);
        double r58124 = r58103 * r58123;
        double r58125 = pow(r58104, r58120);
        double r58126 = r58108 * r58125;
        double r58127 = r58124 / r58126;
        double r58128 = r58122 - r58127;
        double r58129 = r58117 - r58128;
        double r58130 = r58120 * r58105;
        double r58131 = pow(r58109, r58130);
        double r58132 = pow(r58108, r58130);
        double r58133 = r58131 - r58132;
        double r58134 = r58111 + r58110;
        double r58135 = r58133 / r58134;
        double r58136 = r58116 ? r58129 : r58135;
        double r58137 = r58107 ? r58114 : r58136;
        return r58137;
}

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) < -2.4934594501765746e-07

    1. Initial program 1.6

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

    3. Applied add-log-exp1.9

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

    4. Applied add-log-exp1.8

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

    5. Applied diff-log1.8

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

    6. Simplified1.8

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

    if -2.4934594501765746e-07 < (/ 1.0 n) < 1.780710866682253e-05

    1. Initial program 44.8

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

    3. Applied add-log-exp44.8

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

    4. Taylor expanded around inf 32.0

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

    5. Simplified31.4

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

    if 1.780710866682253e-05 < (/ 1.0 n)

    1. Initial program 5.1

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

    3. Applied add-log-exp5.2

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

    5. Applied flip--5.2

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

    6. Simplified5.1

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

    7. Simplified5.1

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

  4. Final simplification23.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.4934594501765746 \cdot 10^{-7}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.780710866682253 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{1 \cdot \log x}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} - {x}^{\left(2 \cdot \frac{1}{n}\right)}}{{x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))