Average Error: 29.4 → 22.7
Time: 12.0s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -934833342441455353856 \lor \neg \left(n \le 59.535473331148146769464801764115691185\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\mathsf{fma}\left(3, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{3}, -{x}^{\left(\frac{1}{n}\right)}\right)\right)}^{3}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -934833342441455353856 \lor \neg \left(n \le 59.535473331148146769464801764115691185\right):\\
\;\;\;\;\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}:\\
\;\;\;\;\sqrt[3]{{\left(\mathsf{fma}\left(3, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{3}, -{x}^{\left(\frac{1}{n}\right)}\right)\right)}^{3}}\\

\end{array}
double f(double x, double n) {
        double r76506 = x;
        double r76507 = 1.0;
        double r76508 = r76506 + r76507;
        double r76509 = n;
        double r76510 = r76507 / r76509;
        double r76511 = pow(r76508, r76510);
        double r76512 = pow(r76506, r76510);
        double r76513 = r76511 - r76512;
        return r76513;
}
↓
double f(double x, double n) {
        double r76514 = n;
        double r76515 = -9.348333424414554e+20;
        bool r76516 = r76514 <= r76515;
        double r76517 = 59.53547333114815;
        bool r76518 = r76514 <= r76517;
        double r76519 = !r76518;
        bool r76520 = r76516 || r76519;
        double r76521 = 1.0;
        double r76522 = 1.0;
        double r76523 = x;
        double r76524 = r76523 * r76514;
        double r76525 = r76522 / r76524;
        double r76526 = 0.5;
        double r76527 = 2.0;
        double r76528 = pow(r76523, r76527);
        double r76529 = r76528 * r76514;
        double r76530 = r76522 / r76529;
        double r76531 = r76522 / r76523;
        double r76532 = log(r76531);
        double r76533 = pow(r76514, r76527);
        double r76534 = r76523 * r76533;
        double r76535 = r76532 / r76534;
        double r76536 = r76521 * r76535;
        double r76537 = fma(r76526, r76530, r76536);
        double r76538 = -r76537;
        double r76539 = fma(r76521, r76525, r76538);
        double r76540 = 3.0;
        double r76541 = r76523 + r76521;
        double r76542 = r76521 / r76514;
        double r76543 = pow(r76541, r76542);
        double r76544 = 0.3333333333333333;
        double r76545 = r76543 * r76544;
        double r76546 = pow(r76523, r76542);
        double r76547 = -r76546;
        double r76548 = fma(r76540, r76545, r76547);
        double r76549 = pow(r76548, r76540);
        double r76550 = cbrt(r76549);
        double r76551 = r76520 ? r76539 : r76550;
        return r76551;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes

  2. if n < -9.348333424414554e+20 or 59.53547333114815 < n

    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 32.8

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

    3. Simplified32.8

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

    if -9.348333424414554e+20 < n < 59.53547333114815

    1. Initial program 9.4

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

    3. Applied add-log-exp9.4

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

    5. Applied add-cube-cbrt9.8

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

    6. Applied log-prod9.9

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

    7. Simplified9.9

      \[\leadsto \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)} + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\]
    8. Using strategy rm

    9. Applied pow1/39.6

      \[\leadsto \left(2 \cdot \log \color{blue}{\left({\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^{\frac{1}{3}}\right)} + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\]
    10. Using strategy rm

    11. Applied add-cbrt-cube9.5

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

    12. Simplified9.4

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

  4. Final simplification22.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -934833342441455353856 \lor \neg \left(n \le 59.535473331148146769464801764115691185\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\mathsf{fma}\left(3, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{3}, -{x}^{\left(\frac{1}{n}\right)}\right)\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))