Average Error: 29.0 → 19.2
Time: 5.6m
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -5.126018534815868 \cdot 10^{+17}:\\ \;\;\;\;(\left(\frac{\frac{-1}{2}}{x}\right) \cdot \left(\frac{1}{x \cdot n}\right) + \left(\frac{1}{x \cdot n} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right))_*\\ \mathbf{elif}\;n \le -2.0474870811742806 \cdot 10^{-307}:\\ \;\;\;\;\left(\log \left(\sqrt[3]{e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}} \cdot \sqrt[3]{e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;n \le 30894828.10545922:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\frac{-1}{2}}{x}\right) \cdot \left(\frac{1}{x \cdot n}\right) + \left(\frac{1}{x \cdot n} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right))_*\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -5.126018534815868 \cdot 10^{+17}:\\
\;\;\;\;(\left(\frac{\frac{-1}{2}}{x}\right) \cdot \left(\frac{1}{x \cdot n}\right) + \left(\frac{1}{x \cdot n} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right))_*\\

\mathbf{elif}\;n \le -2.0474870811742806 \cdot 10^{-307}:\\
\;\;\;\;\left(\log \left(\sqrt[3]{e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}} \cdot \sqrt[3]{e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\

\mathbf{elif}\;n \le 30894828.10545922:\\
\;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\

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

\end{array}
double f(double x, double n) {
        double r29096983 = x;
        double r29096984 = 1.0;
        double r29096985 = r29096983 + r29096984;
        double r29096986 = n;
        double r29096987 = r29096984 / r29096986;
        double r29096988 = pow(r29096985, r29096987);
        double r29096989 = pow(r29096983, r29096987);
        double r29096990 = r29096988 - r29096989;
        return r29096990;
}
↓
double f(double x, double n) {
        double r29096991 = n;
        double r29096992 = -5.126018534815868e+17;
        bool r29096993 = r29096991 <= r29096992;
        double r29096994 = -0.5;
        double r29096995 = x;
        double r29096996 = r29096994 / r29096995;
        double r29096997 = 1.0;
        double r29096998 = r29096995 * r29096991;
        double r29096999 = r29096997 / r29096998;
        double r29097000 = log(r29096995);
        double r29097001 = -r29097000;
        double r29097002 = r29096998 * r29096991;
        double r29097003 = r29097001 / r29097002;
        double r29097004 = r29096999 - r29097003;
        double r29097005 = fma(r29096996, r29096999, r29097004);
        double r29097006 = -2.0474870811742806e-307;
        bool r29097007 = r29096991 <= r29097006;
        double r29097008 = r29096997 + r29096995;
        double r29097009 = r29096997 / r29096991;
        double r29097010 = pow(r29097008, r29097009);
        double r29097011 = pow(r29096995, r29097009);
        double r29097012 = r29097010 - r29097011;
        double r29097013 = cbrt(r29097012);
        double r29097014 = r29097013 * r29097013;
        double r29097015 = exp(r29097014);
        double r29097016 = cbrt(r29097015);
        double r29097017 = r29097016 * r29097016;
        double r29097018 = log(r29097017);
        double r29097019 = log(r29097016);
        double r29097020 = r29097018 + r29097019;
        double r29097021 = r29097020 * r29097013;
        double r29097022 = 30894828.10545922;
        bool r29097023 = r29096991 <= r29097022;
        double r29097024 = log1p(r29096995);
        double r29097025 = r29097024 / r29096991;
        double r29097026 = exp(r29097025);
        double r29097027 = r29097026 - r29097011;
        double r29097028 = r29097023 ? r29097027 : r29097005;
        double r29097029 = r29097007 ? r29097021 : r29097028;
        double r29097030 = r29096993 ? r29097005 : r29097029;
        return r29097030;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if n < -5.126018534815868e+17 or 30894828.10545922 < n

    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 32.6

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

    3. Simplified32.6

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

    if -5.126018534815868e+17 < n < -2.0474870811742806e-307

    1. Initial program 2.1

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

    3. Applied add-log-exp2.2

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

    5. Applied add-cube-cbrt2.2

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

    6. Applied exp-prod2.2

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

    7. Applied log-pow2.2

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

    9. Applied add-cube-cbrt2.2

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

    10. Applied log-prod2.2

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

    if -2.0474870811742806e-307 < n < 30894828.10545922

    1. Initial program 23.3

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

    3. Applied add-exp-log23.6

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

    4. Applied pow-exp23.6

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

    5. Simplified1.7

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

  4. Final simplification19.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -5.126018534815868 \cdot 10^{+17}:\\ \;\;\;\;(\left(\frac{\frac{-1}{2}}{x}\right) \cdot \left(\frac{1}{x \cdot n}\right) + \left(\frac{1}{x \cdot n} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right))_*\\ \mathbf{elif}\;n \le -2.0474870811742806 \cdot 10^{-307}:\\ \;\;\;\;\left(\log \left(\sqrt[3]{e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}} \cdot \sqrt[3]{e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;n \le 30894828.10545922:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\frac{-1}{2}}{x}\right) \cdot \left(\frac{1}{x \cdot n}\right) + \left(\frac{1}{x \cdot n} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))