Average Error: 2.1 → 0.1
Time: 15.3s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 1.299700680114583599056872411063098983944 \cdot 10^{139}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, 10 + k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{k}, \frac{e^{\log k \cdot m}}{k}, \left(\frac{99}{k \cdot k} - \frac{10}{k}\right) \cdot \left(e^{\log k \cdot m} \cdot \frac{a}{k \cdot k}\right)\right)\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 1.299700680114583599056872411063098983944 \cdot 10^{139}:\\
\;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, 10 + k, 1\right)}\\

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

\end{array}
double f(double a, double k, double m) {
        double r222611 = a;
        double r222612 = k;
        double r222613 = m;
        double r222614 = pow(r222612, r222613);
        double r222615 = r222611 * r222614;
        double r222616 = 1.0;
        double r222617 = 10.0;
        double r222618 = r222617 * r222612;
        double r222619 = r222616 + r222618;
        double r222620 = r222612 * r222612;
        double r222621 = r222619 + r222620;
        double r222622 = r222615 / r222621;
        return r222622;
}
↓
double f(double a, double k, double m) {
        double r222623 = k;
        double r222624 = 1.2997006801145836e+139;
        bool r222625 = r222623 <= r222624;
        double r222626 = a;
        double r222627 = m;
        double r222628 = pow(r222623, r222627);
        double r222629 = 10.0;
        double r222630 = r222629 + r222623;
        double r222631 = 1.0;
        double r222632 = fma(r222623, r222630, r222631);
        double r222633 = r222628 / r222632;
        double r222634 = r222626 * r222633;
        double r222635 = r222626 / r222623;
        double r222636 = log(r222623);
        double r222637 = r222636 * r222627;
        double r222638 = exp(r222637);
        double r222639 = r222638 / r222623;
        double r222640 = 99.0;
        double r222641 = r222623 * r222623;
        double r222642 = r222640 / r222641;
        double r222643 = r222629 / r222623;
        double r222644 = r222642 - r222643;
        double r222645 = r222626 / r222641;
        double r222646 = r222638 * r222645;
        double r222647 = r222644 * r222646;
        double r222648 = fma(r222635, r222639, r222647);
        double r222649 = r222625 ? r222634 : r222648;
        return r222649;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 1.2997006801145836e+139

    1. Initial program 0.1

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
    2. Using strategy rm

    3. Applied associate-+l+0.1

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

    4. Simplified0.0

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

    6. Applied *-un-lft-identity0.0

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

    7. Applied times-frac0.1

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

    8. Simplified0.1

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

    9. Simplified0.1

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

    if 1.2997006801145836e+139 < k

    1. Initial program 10.2

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
    2. Using strategy rm

    3. Applied associate-+l+10.2

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

    4. Simplified10.2

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

    5. Taylor expanded around -inf 64.0

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

    6. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a}{k}, \frac{e^{\left(0 + \log k\right) \cdot m}}{k}, \left(\frac{a}{k \cdot k} \cdot e^{\left(0 + \log k\right) \cdot m}\right) \cdot \left(\frac{99}{k \cdot k} - \frac{10}{k}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1.299700680114583599056872411063098983944 \cdot 10^{139}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, 10 + k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{k}, \frac{e^{\log k \cdot m}}{k}, \left(\frac{99}{k \cdot k} - \frac{10}{k}\right) \cdot \left(e^{\log k \cdot m} \cdot \frac{a}{k \cdot k}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))