Average Error: 29.4 → 21.9
Time: 30.8s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.881716821253282740189674058606077167166 \cdot 10^{-9} \lor \neg \left(\frac{1}{n} \le 1.618701588232820408916161347701034856073 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(-\frac{\frac{\log x}{n \cdot n}}{x}, 1, \frac{\frac{0.5}{n}}{{x}^{2}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -1.881716821253282740189674058606077167166 \cdot 10^{-9} \lor \neg \left(\frac{1}{n} \le 1.618701588232820408916161347701034856073 \cdot 10^{-5}\right):\\
\;\;\;\;\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{1}{n}\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\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)\\

\end{array}
double f(double x, double n) {
        double r104148 = x;
        double r104149 = 1.0;
        double r104150 = r104148 + r104149;
        double r104151 = n;
        double r104152 = r104149 / r104151;
        double r104153 = pow(r104150, r104152);
        double r104154 = pow(r104148, r104152);
        double r104155 = r104153 - r104154;
        return r104155;
}
↓
double f(double x, double n) {
        double r104156 = 1.0;
        double r104157 = n;
        double r104158 = r104156 / r104157;
        double r104159 = -1.8817168212532827e-09;
        bool r104160 = r104158 <= r104159;
        double r104161 = 1.6187015882328204e-05;
        bool r104162 = r104158 <= r104161;
        double r104163 = !r104162;
        bool r104164 = r104160 || r104163;
        double r104165 = x;
        double r104166 = r104165 + r104156;
        double r104167 = cbrt(r104166);
        double r104168 = r104167 * r104167;
        double r104169 = pow(r104168, r104158);
        double r104170 = pow(r104167, r104158);
        double r104171 = pow(r104165, r104158);
        double r104172 = -r104171;
        double r104173 = fma(r104169, r104170, r104172);
        double r104174 = r104156 / r104165;
        double r104175 = r104174 / r104157;
        double r104176 = log(r104165);
        double r104177 = r104157 * r104157;
        double r104178 = r104176 / r104177;
        double r104179 = r104178 / r104165;
        double r104180 = -r104179;
        double r104181 = 0.5;
        double r104182 = r104181 / r104157;
        double r104183 = 2.0;
        double r104184 = pow(r104165, r104183);
        double r104185 = r104182 / r104184;
        double r104186 = fma(r104180, r104156, r104185);
        double r104187 = r104175 - r104186;
        double r104188 = r104164 ? r104173 : r104187;
        return r104188;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes

  2. if (/ 1.0 n) < -1.8817168212532827e-09 or 1.6187015882328204e-05 < (/ 1.0 n)

    1. Initial program 8.8

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

    3. Applied add-cube-cbrt8.9

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

    4. Applied unpow-prod-down8.9

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

    5. Applied fma-neg8.9

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

    if -1.8817168212532827e-09 < (/ 1.0 n) < 1.6187015882328204e-05

    1. Initial program 44.9

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

    2. Taylor expanded around inf 32.3

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

    3. Simplified31.6

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(\frac{\frac{-\log x}{n \cdot n}}{x}, 1, \frac{\frac{0.5}{n}}{{x}^{2}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification21.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.881716821253282740189674058606077167166 \cdot 10^{-9} \lor \neg \left(\frac{1}{n} \le 1.618701588232820408916161347701034856073 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(-\frac{\frac{\log x}{n \cdot n}}{x}, 1, \frac{\frac{0.5}{n}}{{x}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))