Average Error: 29.7 → 19.2
Time: 47.0s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -210118.23628303723:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} + \frac{\frac{1}{n}}{x} \cdot \frac{\log x}{n}\right) - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\\ \mathbf{elif}\;n \le -1.1373843048178 \cdot 10^{-311}:\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) + \sqrt[3]{\mathsf{fma}\left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right)\\ \mathbf{elif}\;n \le 94933831758200.45:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} + \frac{\frac{1}{n}}{x} \cdot \frac{\log x}{n}\right) - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -210118.23628303723:\\
\;\;\;\;\left(\frac{\frac{1}{n}}{x} + \frac{\frac{1}{n}}{x} \cdot \frac{\log x}{n}\right) - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\\

\mathbf{elif}\;n \le -1.1373843048178 \cdot 10^{-311}:\\
\;\;\;\;\mathsf{fma}\left({\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) + \sqrt[3]{\mathsf{fma}\left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right)\\

\mathbf{elif}\;n \le 94933831758200.45:\\
\;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\\

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

\end{array}
double f(double x, double n) {
        double r1916096 = x;
        double r1916097 = 1.0;
        double r1916098 = r1916096 + r1916097;
        double r1916099 = n;
        double r1916100 = r1916097 / r1916099;
        double r1916101 = pow(r1916098, r1916100);
        double r1916102 = pow(r1916096, r1916100);
        double r1916103 = r1916101 - r1916102;
        return r1916103;
}
↓
double f(double x, double n) {
        double r1916104 = n;
        double r1916105 = -210118.23628303723;
        bool r1916106 = r1916104 <= r1916105;
        double r1916107 = 1.0;
        double r1916108 = r1916107 / r1916104;
        double r1916109 = x;
        double r1916110 = r1916108 / r1916109;
        double r1916111 = log(r1916109);
        double r1916112 = r1916111 / r1916104;
        double r1916113 = r1916110 * r1916112;
        double r1916114 = r1916110 + r1916113;
        double r1916115 = 0.5;
        double r1916116 = r1916109 * r1916109;
        double r1916117 = r1916115 / r1916116;
        double r1916118 = r1916117 / r1916104;
        double r1916119 = r1916114 - r1916118;
        double r1916120 = -1.1373843048178e-311;
        bool r1916121 = r1916104 <= r1916120;
        double r1916122 = r1916107 + r1916109;
        double r1916123 = cbrt(r1916122);
        double r1916124 = r1916123 * r1916123;
        double r1916125 = pow(r1916124, r1916108);
        double r1916126 = pow(r1916123, r1916108);
        double r1916127 = 2.0;
        double r1916128 = r1916108 / r1916127;
        double r1916129 = pow(r1916109, r1916128);
        double r1916130 = r1916129 * r1916129;
        double r1916131 = -r1916130;
        double r1916132 = fma(r1916125, r1916126, r1916131);
        double r1916133 = -r1916129;
        double r1916134 = fma(r1916133, r1916129, r1916130);
        double r1916135 = cbrt(r1916134);
        double r1916136 = r1916135 * r1916135;
        double r1916137 = r1916135 * r1916136;
        double r1916138 = r1916132 + r1916137;
        double r1916139 = 94933831758200.45;
        bool r1916140 = r1916104 <= r1916139;
        double r1916141 = log1p(r1916109);
        double r1916142 = r1916141 / r1916104;
        double r1916143 = exp(r1916142);
        double r1916144 = pow(r1916109, r1916108);
        double r1916145 = log1p(r1916144);
        double r1916146 = expm1(r1916145);
        double r1916147 = r1916143 - r1916146;
        double r1916148 = r1916140 ? r1916147 : r1916119;
        double r1916149 = r1916121 ? r1916138 : r1916148;
        double r1916150 = r1916106 ? r1916119 : r1916149;
        return r1916150;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if n < -210118.23628303723 or 94933831758200.45 < n

    1. Initial program 45.2

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

    3. Applied expm1-log1p-u45.2

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)}\]

    4. Taylor expanded around inf 32.7

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

    5. Simplified32.2

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

    if -210118.23628303723 < n < -1.1373843048178e-311

    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 sqr-pow0.6

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

    4. Applied add-cube-cbrt0.6

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

    5. Applied unpow-prod-down0.6

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

    6. Applied prod-diff0.6

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

    8. Applied add-cube-cbrt0.6

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

    if -1.1373843048178e-311 < n < 94933831758200.45

    1. Initial program 26.3

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

    3. Applied expm1-log1p-u26.3

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)}\]
    4. Using strategy rm

    5. Applied pow-to-exp26.3

      \[\leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\]

    6. Simplified3.7

      \[\leadsto e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification19.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -210118.23628303723:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} + \frac{\frac{1}{n}}{x} \cdot \frac{\log x}{n}\right) - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\\ \mathbf{elif}\;n \le -1.1373843048178 \cdot 10^{-311}:\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) + \sqrt[3]{\mathsf{fma}\left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right)\\ \mathbf{elif}\;n \le 94933831758200.45:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} + \frac{\frac{1}{n}}{x} \cdot \frac{\log x}{n}\right) - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))