Average Error: 33.0 → 25.0
Time: 13.3s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.6345588493135571 \cdot 10^{-23}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 5.53064647223408922 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-{x}^{\left(\frac{1}{n}\right)}, {x}^{\left(2 \cdot \frac{1}{n}\right)}, {\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}{\mathsf{fma}\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}, {x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -1.6345588493135571 \cdot 10^{-23}:\\
\;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\

\mathbf{elif}\;\frac{1}{n} \le 5.53064647223408922 \cdot 10^{-24}:\\
\;\;\;\;\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}:\\
\;\;\;\;\frac{\mathsf{fma}\left(-{x}^{\left(\frac{1}{n}\right)}, {x}^{\left(2 \cdot \frac{1}{n}\right)}, {\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}{\mathsf{fma}\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}, {x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}\\

\end{array}
double f(double x, double n) {
        double r54376 = x;
        double r54377 = 1.0;
        double r54378 = r54376 + r54377;
        double r54379 = n;
        double r54380 = r54377 / r54379;
        double r54381 = pow(r54378, r54380);
        double r54382 = pow(r54376, r54380);
        double r54383 = r54381 - r54382;
        return r54383;
}
↓
double f(double x, double n) {
        double r54384 = 1.0;
        double r54385 = n;
        double r54386 = r54384 / r54385;
        double r54387 = -1.634558849313557e-23;
        bool r54388 = r54386 <= r54387;
        double r54389 = x;
        double r54390 = r54389 + r54384;
        double r54391 = pow(r54390, r54386);
        double r54392 = pow(r54389, r54386);
        double r54393 = r54391 - r54392;
        double r54394 = exp(r54393);
        double r54395 = log(r54394);
        double r54396 = 5.530646472234089e-24;
        bool r54397 = r54386 <= r54396;
        double r54398 = 1.0;
        double r54399 = r54389 * r54385;
        double r54400 = r54398 / r54399;
        double r54401 = 0.5;
        double r54402 = 2.0;
        double r54403 = pow(r54389, r54402);
        double r54404 = r54403 * r54385;
        double r54405 = r54398 / r54404;
        double r54406 = r54398 / r54389;
        double r54407 = log(r54406);
        double r54408 = pow(r54385, r54402);
        double r54409 = r54389 * r54408;
        double r54410 = r54407 / r54409;
        double r54411 = r54384 * r54410;
        double r54412 = fma(r54401, r54405, r54411);
        double r54413 = -r54412;
        double r54414 = fma(r54384, r54400, r54413);
        double r54415 = -r54392;
        double r54416 = r54402 * r54386;
        double r54417 = pow(r54389, r54416);
        double r54418 = 3.0;
        double r54419 = pow(r54391, r54418);
        double r54420 = fma(r54415, r54417, r54419);
        double r54421 = r54391 + r54392;
        double r54422 = pow(r54390, r54416);
        double r54423 = fma(r54421, r54392, r54422);
        double r54424 = r54420 / r54423;
        double r54425 = r54397 ? r54414 : r54424;
        double r54426 = r54388 ? r54395 : r54425;
        return r54426;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (/ 1.0 n) < -1.634558849313557e-23

    1. Initial program 5.5

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

    3. Applied add-log-exp5.9

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

    4. Applied add-log-exp5.8

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

    5. Applied diff-log5.8

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

    6. Simplified5.8

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

    if -1.634558849313557e-23 < (/ 1.0 n) < 5.530646472234089e-24

    1. Initial program 44.6

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

    2. Taylor expanded around inf 32.6

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

    3. Simplified32.6

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

    if 5.530646472234089e-24 < (/ 1.0 n)

    1. Initial program 13.3

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

    3. Applied sqr-pow13.4

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

    5. Applied flip3--13.5

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

    6. Simplified13.5

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

    7. Simplified13.5

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

  4. Final simplification25.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.6345588493135571 \cdot 10^{-23}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 5.53064647223408922 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-{x}^{\left(\frac{1}{n}\right)}, {x}^{\left(2 \cdot \frac{1}{n}\right)}, {\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}{\mathsf{fma}\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}, {x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))