Average Error: 29.3 → 22.2
Time: 30.2s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.140830927760539067774892796574941200571 \cdot 10^{-9}:\\ \;\;\;\;\log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.080776736319869253764442313662561201966 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(-\frac{\frac{\log x}{n \cdot n}}{x}, 1, \frac{\frac{0.5}{n}}{{x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -1.140830927760539067774892796574941200571 \cdot 10^{-9}:\\
\;\;\;\;\log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\

\mathbf{elif}\;\frac{1}{n} \le 1.080776736319869253764442313662561201966 \cdot 10^{-26}:\\
\;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(-\frac{\frac{\log x}{n \cdot n}}{x}, 1, \frac{\frac{0.5}{n}}{{x}^{2}}\right)\\

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

\end{array}
double f(double x, double n) {
        double r75560 = x;
        double r75561 = 1.0;
        double r75562 = r75560 + r75561;
        double r75563 = n;
        double r75564 = r75561 / r75563;
        double r75565 = pow(r75562, r75564);
        double r75566 = pow(r75560, r75564);
        double r75567 = r75565 - r75566;
        return r75567;
}
↓
double f(double x, double n) {
        double r75568 = 1.0;
        double r75569 = n;
        double r75570 = r75568 / r75569;
        double r75571 = -1.140830927760539e-09;
        bool r75572 = r75570 <= r75571;
        double r75573 = x;
        double r75574 = r75573 + r75568;
        double r75575 = pow(r75574, r75570);
        double r75576 = pow(r75573, r75570);
        double r75577 = r75575 - r75576;
        double r75578 = exp(r75577);
        double r75579 = sqrt(r75578);
        double r75580 = log(r75579);
        double r75581 = r75580 + r75580;
        double r75582 = 1.0807767363198693e-26;
        bool r75583 = r75570 <= r75582;
        double r75584 = r75568 / r75573;
        double r75585 = r75584 / r75569;
        double r75586 = log(r75573);
        double r75587 = r75569 * r75569;
        double r75588 = r75586 / r75587;
        double r75589 = r75588 / r75573;
        double r75590 = -r75589;
        double r75591 = 0.5;
        double r75592 = r75591 / r75569;
        double r75593 = 2.0;
        double r75594 = pow(r75573, r75593);
        double r75595 = r75592 / r75594;
        double r75596 = fma(r75590, r75568, r75595);
        double r75597 = r75585 - r75596;
        double r75598 = log(r75578);
        double r75599 = r75583 ? r75597 : r75598;
        double r75600 = r75572 ? r75581 : r75599;
        return r75600;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (/ 1.0 n) < -1.140830927760539e-09

    1. Initial program 0.7

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

    3. Applied add-log-exp0.9

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

    4. Applied add-log-exp0.9

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

    5. Applied diff-log0.9

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

    6. Simplified0.9

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

    8. Applied add-sqr-sqrt0.9

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

    9. Applied log-prod0.9

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

    if -1.140830927760539e-09 < (/ 1.0 n) < 1.0807767363198693e-26

    1. Initial program 44.7

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

    2. Taylor expanded around inf 32.4

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

    3. Simplified31.8

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

    if 1.0807767363198693e-26 < (/ 1.0 n)

    1. Initial program 27.6

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

    3. Applied add-log-exp27.7

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

    4. Applied add-log-exp27.7

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

    5. Applied diff-log27.7

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

    6. Simplified27.7

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

  4. Final simplification22.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.140830927760539067774892796574941200571 \cdot 10^{-9}:\\ \;\;\;\;\log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.080776736319869253764442313662561201966 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(-\frac{\frac{\log x}{n \cdot n}}{x}, 1, \frac{\frac{0.5}{n}}{{x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))