Average Error: 29.4 → 22.7
Time: 11.8s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.032379292376500446:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.50994306778113202 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -0.032379292376500446:\\
\;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\\

\mathbf{elif}\;\frac{1}{n} \le 1.50994306778113202 \cdot 10^{-13}:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{e^{\log \left({\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 r56269 = x;
        double r56270 = 1.0;
        double r56271 = r56269 + r56270;
        double r56272 = n;
        double r56273 = r56270 / r56272;
        double r56274 = pow(r56271, r56273);
        double r56275 = pow(r56269, r56273);
        double r56276 = r56274 - r56275;
        return r56276;
}
↓
double f(double x, double n) {
        double r56277 = 1.0;
        double r56278 = n;
        double r56279 = r56277 / r56278;
        double r56280 = -0.032379292376500446;
        bool r56281 = r56279 <= r56280;
        double r56282 = x;
        double r56283 = r56282 + r56277;
        double r56284 = pow(r56283, r56279);
        double r56285 = pow(r56282, r56279);
        double r56286 = log1p(r56285);
        double r56287 = expm1(r56286);
        double r56288 = r56284 - r56287;
        double r56289 = 1.509943067781132e-13;
        bool r56290 = r56279 <= r56289;
        double r56291 = 1.0;
        double r56292 = r56282 * r56278;
        double r56293 = r56291 / r56292;
        double r56294 = 0.5;
        double r56295 = 2.0;
        double r56296 = pow(r56282, r56295);
        double r56297 = r56296 * r56278;
        double r56298 = r56291 / r56297;
        double r56299 = r56291 / r56282;
        double r56300 = log(r56299);
        double r56301 = pow(r56278, r56295);
        double r56302 = r56282 * r56301;
        double r56303 = r56300 / r56302;
        double r56304 = r56277 * r56303;
        double r56305 = fma(r56294, r56298, r56304);
        double r56306 = -r56305;
        double r56307 = fma(r56277, r56293, r56306);
        double r56308 = r56284 - r56285;
        double r56309 = cbrt(r56308);
        double r56310 = exp(r56308);
        double r56311 = log(r56310);
        double r56312 = cbrt(r56311);
        double r56313 = r56309 * r56312;
        double r56314 = log(r56308);
        double r56315 = exp(r56314);
        double r56316 = cbrt(r56315);
        double r56317 = r56313 * r56316;
        double r56318 = r56290 ? r56307 : r56317;
        double r56319 = r56281 ? r56288 : r56318;
        return r56319;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (/ 1.0 n) < -0.032379292376500446

    1. Initial program 0.1

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

    3. Applied expm1-log1p-u0.1

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

    if -0.032379292376500446 < (/ 1.0 n) < 1.509943067781132e-13

    1. Initial program 44.4

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

    2. Taylor expanded around inf 32.9

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

    3. Simplified32.9

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

    if 1.509943067781132e-13 < (/ 1.0 n)

    1. Initial program 24.5

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

    3. Applied add-cube-cbrt24.5

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

    5. Applied add-log-exp24.6

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

    6. Applied add-log-exp24.7

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

    7. Applied diff-log24.7

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

    8. Simplified24.6

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

    10. Applied add-exp-log24.6

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

  4. Final simplification22.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.032379292376500446:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.50994306778113202 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))