Average Error: 2.2 → 0.1
Time: 14.1s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 8.14236707063809 \cdot 10^{+129}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k + 10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{\log k \cdot m}}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{a}}, 99, \frac{a}{k} \cdot \frac{e^{\log k \cdot m}}{k} - \frac{\left(10 \cdot a\right) \cdot e^{\log k \cdot m}}{k \cdot \left(k \cdot k\right)}\right)\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 8.14236707063809 \cdot 10^{+129}:\\
\;\;\;\;\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k + 10, k, 1\right)}\\

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

\end{array}
double f(double a, double k, double m) {
        double r7417601 = a;
        double r7417602 = k;
        double r7417603 = m;
        double r7417604 = pow(r7417602, r7417603);
        double r7417605 = r7417601 * r7417604;
        double r7417606 = 1.0;
        double r7417607 = 10.0;
        double r7417608 = r7417607 * r7417602;
        double r7417609 = r7417606 + r7417608;
        double r7417610 = r7417602 * r7417602;
        double r7417611 = r7417609 + r7417610;
        double r7417612 = r7417605 / r7417611;
        return r7417612;
}
↓
double f(double a, double k, double m) {
        double r7417613 = k;
        double r7417614 = 8.14236707063809e+129;
        bool r7417615 = r7417613 <= r7417614;
        double r7417616 = m;
        double r7417617 = pow(r7417613, r7417616);
        double r7417618 = a;
        double r7417619 = r7417617 * r7417618;
        double r7417620 = 10.0;
        double r7417621 = r7417613 + r7417620;
        double r7417622 = 1.0;
        double r7417623 = fma(r7417621, r7417613, r7417622);
        double r7417624 = r7417619 / r7417623;
        double r7417625 = log(r7417613);
        double r7417626 = r7417625 * r7417616;
        double r7417627 = exp(r7417626);
        double r7417628 = r7417613 * r7417613;
        double r7417629 = r7417628 * r7417628;
        double r7417630 = r7417629 / r7417618;
        double r7417631 = r7417627 / r7417630;
        double r7417632 = 99.0;
        double r7417633 = r7417618 / r7417613;
        double r7417634 = r7417627 / r7417613;
        double r7417635 = r7417633 * r7417634;
        double r7417636 = r7417620 * r7417618;
        double r7417637 = r7417636 * r7417627;
        double r7417638 = r7417613 * r7417628;
        double r7417639 = r7417637 / r7417638;
        double r7417640 = r7417635 - r7417639;
        double r7417641 = fma(r7417631, r7417632, r7417640);
        double r7417642 = r7417615 ? r7417624 : r7417641;
        return r7417642;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 8.14236707063809e+129

    1. Initial program 0.1

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

    2. Simplified0.1

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

    if 8.14236707063809e+129 < k

    1. Initial program 9.7

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

    2. Simplified9.7

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

    4. Applied add-sqr-sqrt9.7

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

    5. Applied times-frac9.7

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

    6. Taylor expanded around inf 9.7

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

    7. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 8.14236707063809 \cdot 10^{+129}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k + 10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{\log k \cdot m}}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{a}}, 99, \frac{a}{k} \cdot \frac{e^{\log k \cdot m}}{k} - \frac{\left(10 \cdot a\right) \cdot e^{\log k \cdot m}}{k \cdot \left(k \cdot k\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))