Average Error: 29.2 → 18.9
Time: 34.8s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -30729818.785387494:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x \cdot \left(x \cdot n\right)}, \frac{-1}{2}, \frac{\frac{1}{x}}{n} + \frac{\frac{\log x}{n}}{x \cdot n}\right) + \mathsf{fma}\left(-{x}^{\left(\frac{1}{n}\right)}, 1, {x}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;n \le -6.7690117056222 \cdot 10^{-311}:\\ \;\;\;\;\mathsf{fma}\left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}, \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) + \mathsf{fma}\left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}, \left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)\\ \mathbf{elif}\;n \le 2.9397635619222146 \cdot 10^{+20}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\log x}{n \cdot n}}{x} + \frac{\frac{1}{n}}{x}\right) - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -30729818.785387494:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{x \cdot \left(x \cdot n\right)}, \frac{-1}{2}, \frac{\frac{1}{x}}{n} + \frac{\frac{\log x}{n}}{x \cdot n}\right) + \mathsf{fma}\left(-{x}^{\left(\frac{1}{n}\right)}, 1, {x}^{\left(\frac{1}{n}\right)}\right)\\

\mathbf{elif}\;n \le -6.7690117056222 \cdot 10^{-311}:\\
\;\;\;\;\mathsf{fma}\left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}, \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) + \mathsf{fma}\left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}, \left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)\\

\mathbf{elif}\;n \le 2.9397635619222146 \cdot 10^{+20}:\\
\;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\

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

\end{array}
double f(double x, double n) {
        double r2281718 = x;
        double r2281719 = 1.0;
        double r2281720 = r2281718 + r2281719;
        double r2281721 = n;
        double r2281722 = r2281719 / r2281721;
        double r2281723 = pow(r2281720, r2281722);
        double r2281724 = pow(r2281718, r2281722);
        double r2281725 = r2281723 - r2281724;
        return r2281725;
}
↓
double f(double x, double n) {
        double r2281726 = n;
        double r2281727 = -30729818.785387494;
        bool r2281728 = r2281726 <= r2281727;
        double r2281729 = 1.0;
        double r2281730 = x;
        double r2281731 = r2281730 * r2281726;
        double r2281732 = r2281730 * r2281731;
        double r2281733 = r2281729 / r2281732;
        double r2281734 = -0.5;
        double r2281735 = r2281729 / r2281730;
        double r2281736 = r2281735 / r2281726;
        double r2281737 = log(r2281730);
        double r2281738 = r2281737 / r2281726;
        double r2281739 = r2281738 / r2281731;
        double r2281740 = r2281736 + r2281739;
        double r2281741 = fma(r2281733, r2281734, r2281740);
        double r2281742 = r2281729 / r2281726;
        double r2281743 = pow(r2281730, r2281742);
        double r2281744 = -r2281743;
        double r2281745 = fma(r2281744, r2281729, r2281743);
        double r2281746 = r2281741 + r2281745;
        double r2281747 = -6.7690117056222e-311;
        bool r2281748 = r2281726 <= r2281747;
        double r2281749 = cbrt(r2281743);
        double r2281750 = -r2281749;
        double r2281751 = r2281749 * r2281749;
        double r2281752 = r2281751 * r2281749;
        double r2281753 = fma(r2281750, r2281751, r2281752);
        double r2281754 = r2281729 + r2281730;
        double r2281755 = pow(r2281754, r2281742);
        double r2281756 = cbrt(r2281755);
        double r2281757 = r2281756 * r2281756;
        double r2281758 = r2281750 * r2281751;
        double r2281759 = fma(r2281757, r2281756, r2281758);
        double r2281760 = r2281753 + r2281759;
        double r2281761 = 2.9397635619222146e+20;
        bool r2281762 = r2281726 <= r2281761;
        double r2281763 = log1p(r2281730);
        double r2281764 = r2281763 / r2281726;
        double r2281765 = exp(r2281764);
        double r2281766 = r2281765 - r2281743;
        double r2281767 = r2281726 * r2281726;
        double r2281768 = r2281737 / r2281767;
        double r2281769 = r2281768 / r2281730;
        double r2281770 = r2281742 / r2281730;
        double r2281771 = r2281769 + r2281770;
        double r2281772 = 0.5;
        double r2281773 = r2281772 / r2281726;
        double r2281774 = r2281730 * r2281730;
        double r2281775 = r2281773 / r2281774;
        double r2281776 = r2281771 - r2281775;
        double r2281777 = r2281762 ? r2281766 : r2281776;
        double r2281778 = r2281748 ? r2281760 : r2281777;
        double r2281779 = r2281728 ? r2281746 : r2281778;
        return r2281779;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 4 regimes

  2. if n < -30729818.785387494

    1. Initial program 44.2

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

    3. Applied *-un-lft-identity44.2

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

    4. Applied add-sqr-sqrt44.2

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

    5. Applied prod-diff44.3

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

    7. Applied add-log-exp44.2

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

    8. Taylor expanded around inf 32.6

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

    9. Simplified32.0

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

    if -30729818.785387494 < n < -6.7690117056222e-311

    1. Initial program 0.8

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

    3. Applied add-cube-cbrt0.8

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

    4. Applied add-cube-cbrt0.8

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

    5. Applied prod-diff0.8

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

    if -6.7690117056222e-311 < n < 2.9397635619222146e+20

    1. Initial program 28.1

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

    3. Applied pow-to-exp28.1

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

    4. Simplified5.6

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

    if 2.9397635619222146e+20 < n

    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 32.2

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

    3. Simplified31.8

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{n}}{x} + \frac{\frac{\log x}{n \cdot n}}{x}\right) - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification18.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -30729818.785387494:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x \cdot \left(x \cdot n\right)}, \frac{-1}{2}, \frac{\frac{1}{x}}{n} + \frac{\frac{\log x}{n}}{x \cdot n}\right) + \mathsf{fma}\left(-{x}^{\left(\frac{1}{n}\right)}, 1, {x}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;n \le -6.7690117056222 \cdot 10^{-311}:\\ \;\;\;\;\mathsf{fma}\left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}, \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) + \mathsf{fma}\left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}, \left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)\\ \mathbf{elif}\;n \le 2.9397635619222146 \cdot 10^{+20}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\log x}{n \cdot n}}{x} + \frac{\frac{1}{n}}{x}\right) - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))