Average Error: 2.0 → 0.1
Time: 16.7s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 845009784.1006279:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}}{\mathsf{fma}\left(k, k + 10, 1\right) \cdot \mathsf{fma}\left(k, k + 10, 1\right)}} \cdot \left({k}^{m} \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(99, \frac{e^{m \cdot \log k}}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{a}}, \frac{e^{m \cdot \log k}}{k} \cdot \frac{a}{k} - \frac{e^{m \cdot \log k} \cdot 10}{\frac{k \cdot \left(k \cdot k\right)}{a}}\right)\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 845009784.1006279:\\
\;\;\;\;\sqrt[3]{\frac{\frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}}{\mathsf{fma}\left(k, k + 10, 1\right) \cdot \mathsf{fma}\left(k, k + 10, 1\right)}} \cdot \left({k}^{m} \cdot a\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(99, \frac{e^{m \cdot \log k}}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{a}}, \frac{e^{m \cdot \log k}}{k} \cdot \frac{a}{k} - \frac{e^{m \cdot \log k} \cdot 10}{\frac{k \cdot \left(k \cdot k\right)}{a}}\right)\\

\end{array}
double f(double a, double k, double m) {
        double r3745608 = a;
        double r3745609 = k;
        double r3745610 = m;
        double r3745611 = pow(r3745609, r3745610);
        double r3745612 = r3745608 * r3745611;
        double r3745613 = 1.0;
        double r3745614 = 10.0;
        double r3745615 = r3745614 * r3745609;
        double r3745616 = r3745613 + r3745615;
        double r3745617 = r3745609 * r3745609;
        double r3745618 = r3745616 + r3745617;
        double r3745619 = r3745612 / r3745618;
        return r3745619;
}
↓
double f(double a, double k, double m) {
        double r3745620 = k;
        double r3745621 = 845009784.1006279;
        bool r3745622 = r3745620 <= r3745621;
        double r3745623 = 1.0;
        double r3745624 = 10.0;
        double r3745625 = r3745620 + r3745624;
        double r3745626 = fma(r3745620, r3745625, r3745623);
        double r3745627 = r3745623 / r3745626;
        double r3745628 = r3745626 * r3745626;
        double r3745629 = r3745627 / r3745628;
        double r3745630 = cbrt(r3745629);
        double r3745631 = m;
        double r3745632 = pow(r3745620, r3745631);
        double r3745633 = a;
        double r3745634 = r3745632 * r3745633;
        double r3745635 = r3745630 * r3745634;
        double r3745636 = 99.0;
        double r3745637 = log(r3745620);
        double r3745638 = r3745631 * r3745637;
        double r3745639 = exp(r3745638);
        double r3745640 = r3745620 * r3745620;
        double r3745641 = r3745640 * r3745640;
        double r3745642 = r3745641 / r3745633;
        double r3745643 = r3745639 / r3745642;
        double r3745644 = r3745639 / r3745620;
        double r3745645 = r3745633 / r3745620;
        double r3745646 = r3745644 * r3745645;
        double r3745647 = r3745639 * r3745624;
        double r3745648 = r3745620 * r3745640;
        double r3745649 = r3745648 / r3745633;
        double r3745650 = r3745647 / r3745649;
        double r3745651 = r3745646 - r3745650;
        double r3745652 = fma(r3745636, r3745643, r3745651);
        double r3745653 = r3745622 ? r3745635 : r3745652;
        return r3745653;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 845009784.1006279

    1. Initial program 0.1

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k + 10, k, 1\right)}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt0.2

      \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{\sqrt{\mathsf{fma}\left(k + 10, k, 1\right)} \cdot \sqrt{\mathsf{fma}\left(k + 10, k, 1\right)}}}\]

    5. Applied associate-/r*0.2

      \[\leadsto \color{blue}{\frac{\frac{{k}^{m} \cdot a}{\sqrt{\mathsf{fma}\left(k + 10, k, 1\right)}}}{\sqrt{\mathsf{fma}\left(k + 10, k, 1\right)}}}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity0.2

      \[\leadsto \frac{\frac{{k}^{m} \cdot a}{\sqrt{\mathsf{fma}\left(k + 10, k, 1\right)}}}{\sqrt{\color{blue}{1 \cdot \mathsf{fma}\left(k + 10, k, 1\right)}}}\]

    8. Applied sqrt-prod0.2

      \[\leadsto \frac{\frac{{k}^{m} \cdot a}{\sqrt{\mathsf{fma}\left(k + 10, k, 1\right)}}}{\color{blue}{\sqrt{1} \cdot \sqrt{\mathsf{fma}\left(k + 10, k, 1\right)}}}\]

    9. Applied div-inv0.2

      \[\leadsto \frac{\color{blue}{\left({k}^{m} \cdot a\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(k + 10, k, 1\right)}}}}{\sqrt{1} \cdot \sqrt{\mathsf{fma}\left(k + 10, k, 1\right)}}\]

    10. Applied times-frac0.2

      \[\leadsto \color{blue}{\frac{{k}^{m} \cdot a}{\sqrt{1}} \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(k + 10, k, 1\right)}}}{\sqrt{\mathsf{fma}\left(k + 10, k, 1\right)}}}\]

    11. Simplified0.2

      \[\leadsto \color{blue}{\left({k}^{m} \cdot a\right)} \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(k + 10, k, 1\right)}}}{\sqrt{\mathsf{fma}\left(k + 10, k, 1\right)}}\]

    12. Simplified0.0

      \[\leadsto \left({k}^{m} \cdot a\right) \cdot \color{blue}{\frac{1}{\mathsf{fma}\left(10 + k, k, 1\right)}}\]
    13. Using strategy rm

    14. Applied add-cbrt-cube0.0

      \[\leadsto \left({k}^{m} \cdot a\right) \cdot \frac{1}{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(10 + k, k, 1\right) \cdot \mathsf{fma}\left(10 + k, k, 1\right)\right) \cdot \mathsf{fma}\left(10 + k, k, 1\right)}}}\]

    15. Applied add-cbrt-cube0.0

      \[\leadsto \left({k}^{m} \cdot a\right) \cdot \frac{\color{blue}{\sqrt[3]{\left(1 \cdot 1\right) \cdot 1}}}{\sqrt[3]{\left(\mathsf{fma}\left(10 + k, k, 1\right) \cdot \mathsf{fma}\left(10 + k, k, 1\right)\right) \cdot \mathsf{fma}\left(10 + k, k, 1\right)}}\]

    16. Applied cbrt-undiv0.0

      \[\leadsto \left({k}^{m} \cdot a\right) \cdot \color{blue}{\sqrt[3]{\frac{\left(1 \cdot 1\right) \cdot 1}{\left(\mathsf{fma}\left(10 + k, k, 1\right) \cdot \mathsf{fma}\left(10 + k, k, 1\right)\right) \cdot \mathsf{fma}\left(10 + k, k, 1\right)}}}\]

    17. Simplified0.0

      \[\leadsto \left({k}^{m} \cdot a\right) \cdot \sqrt[3]{\color{blue}{\frac{\frac{1}{\mathsf{fma}\left(k, 10 + k, 1\right)}}{\mathsf{fma}\left(k, 10 + k, 1\right) \cdot \mathsf{fma}\left(k, 10 + k, 1\right)}}}\]

    if 845009784.1006279 < k

    1. Initial program 5.2

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]

    2. Simplified5.2

      \[\leadsto \color{blue}{\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k + 10, k, 1\right)}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt5.2

      \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{\sqrt{\mathsf{fma}\left(k + 10, k, 1\right)} \cdot \sqrt{\mathsf{fma}\left(k + 10, k, 1\right)}}}\]

    5. Applied associate-/r*5.2

      \[\leadsto \color{blue}{\frac{\frac{{k}^{m} \cdot a}{\sqrt{\mathsf{fma}\left(k + 10, k, 1\right)}}}{\sqrt{\mathsf{fma}\left(k + 10, k, 1\right)}}}\]

    6. Taylor expanded around inf 5.2

      \[\leadsto \color{blue}{\left(99 \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{4}} + \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{2}}\right) - 10 \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{3}}}\]

    7. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(99, \frac{e^{-\left(-m \cdot \log k\right)}}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{a}}, \frac{a}{k} \cdot \frac{e^{-\left(-m \cdot \log k\right)}}{k} - \frac{e^{-\left(-m \cdot \log k\right)} \cdot 10}{\frac{\left(k \cdot k\right) \cdot k}{a}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 845009784.1006279:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}}{\mathsf{fma}\left(k, k + 10, 1\right) \cdot \mathsf{fma}\left(k, k + 10, 1\right)}} \cdot \left({k}^{m} \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(99, \frac{e^{m \cdot \log k}}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{a}}, \frac{e^{m \cdot \log k}}{k} \cdot \frac{a}{k} - \frac{e^{m \cdot \log k} \cdot 10}{\frac{k \cdot \left(k \cdot k\right)}{a}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))