Average Error: 29.2 → 23.5
Time: 30.5s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.709194808651584388774143866756237940263 \cdot 10^{-10}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) + \mathsf{fma}\left(-\sqrt[3]{{\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}, {x}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.064370619830308532575078878811982196745 \cdot 10^{-50}:\\ \;\;\;\;\frac{1}{x \cdot n} - \mathsf{fma}\left(1, \frac{-\log x}{x \cdot {n}^{2}}, \frac{0.5}{{x}^{2} \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} - {x}^{\left(2 \cdot \frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -1.709194808651584388774143866756237940263 \cdot 10^{-10}:\\
\;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) + \mathsf{fma}\left(-\sqrt[3]{{\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}, {x}^{\left(\frac{1}{n}\right)}\right)\\

\mathbf{elif}\;\frac{1}{n} \le 1.064370619830308532575078878811982196745 \cdot 10^{-50}:\\
\;\;\;\;\frac{1}{x \cdot n} - \mathsf{fma}\left(1, \frac{-\log x}{x \cdot {n}^{2}}, \frac{0.5}{{x}^{2} \cdot n}\right)\\

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

\end{array}
double f(double x, double n) {
        double r66104 = x;
        double r66105 = 1.0;
        double r66106 = r66104 + r66105;
        double r66107 = n;
        double r66108 = r66105 / r66107;
        double r66109 = pow(r66106, r66108);
        double r66110 = pow(r66104, r66108);
        double r66111 = r66109 - r66110;
        return r66111;
}
↓
double f(double x, double n) {
        double r66112 = 1.0;
        double r66113 = n;
        double r66114 = r66112 / r66113;
        double r66115 = -1.7091948086515844e-10;
        bool r66116 = r66114 <= r66115;
        double r66117 = x;
        double r66118 = r66117 + r66112;
        double r66119 = pow(r66118, r66114);
        double r66120 = pow(r66117, r66114);
        double r66121 = r66119 - r66120;
        double r66122 = 2.0;
        double r66123 = pow(r66117, r66122);
        double r66124 = pow(r66123, r66114);
        double r66125 = cbrt(r66124);
        double r66126 = -r66125;
        double r66127 = cbrt(r66120);
        double r66128 = fma(r66126, r66127, r66120);
        double r66129 = r66121 + r66128;
        double r66130 = 1.0643706198303085e-50;
        bool r66131 = r66114 <= r66130;
        double r66132 = r66117 * r66113;
        double r66133 = r66112 / r66132;
        double r66134 = log(r66117);
        double r66135 = -r66134;
        double r66136 = pow(r66113, r66122);
        double r66137 = r66117 * r66136;
        double r66138 = r66135 / r66137;
        double r66139 = 0.5;
        double r66140 = r66123 * r66113;
        double r66141 = r66139 / r66140;
        double r66142 = fma(r66112, r66138, r66141);
        double r66143 = r66133 - r66142;
        double r66144 = r66122 * r66114;
        double r66145 = pow(r66118, r66144);
        double r66146 = pow(r66117, r66144);
        double r66147 = r66145 - r66146;
        double r66148 = r66119 + r66120;
        double r66149 = r66147 / r66148;
        double r66150 = r66131 ? r66143 : r66149;
        double r66151 = r66116 ? r66129 : r66150;
        return r66151;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (/ 1.0 n) < -1.7091948086515844e-10

    1. Initial program 1.2

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

    3. Applied add-cbrt-cube1.4

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

    4. Simplified1.4

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

    6. Applied cube-mult1.4

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

    7. Applied cbrt-prod1.4

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

    8. Applied *-un-lft-identity1.4

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

    9. Applied unpow-prod-down1.4

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

    10. Applied prod-diff1.4

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

    11. Simplified1.3

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

    12. Simplified1.4

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) + \color{blue}{\mathsf{fma}\left(-\sqrt[3]{{x}^{\left(2 \cdot \frac{1}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}, {x}^{\left(\frac{1}{n}\right)}\right)}\]
    13. Using strategy rm

    14. Applied pow-unpow1.9

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) + \mathsf{fma}\left(-\sqrt[3]{\color{blue}{{\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)}}}, \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}, {x}^{\left(\frac{1}{n}\right)}\right)\]

    if -1.7091948086515844e-10 < (/ 1.0 n) < 1.0643706198303085e-50

    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 32.7

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

    3. Simplified32.7

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

    if 1.0643706198303085e-50 < (/ 1.0 n)

    1. Initial program 30.7

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

    3. Applied flip--33.9

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

    4. Simplified33.9

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

  4. Final simplification23.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.709194808651584388774143866756237940263 \cdot 10^{-10}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) + \mathsf{fma}\left(-\sqrt[3]{{\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}, {x}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.064370619830308532575078878811982196745 \cdot 10^{-50}:\\ \;\;\;\;\frac{1}{x \cdot n} - \mathsf{fma}\left(1, \frac{-\log x}{x \cdot {n}^{2}}, \frac{0.5}{{x}^{2} \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} - {x}^{\left(2 \cdot \frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))