Average Error: 2.0 → 0.1
Time: 37.5s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 3.333212644978590011114258413451524263306 \cdot 10^{107}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{99}{k \cdot k}, \frac{e^{m \cdot \log k} \cdot \frac{a}{k}}{k}, \mathsf{fma}\left(\frac{\frac{a}{k}}{\sqrt{k}}, \frac{e^{m \cdot \log k}}{\sqrt{k}}, \left(-\frac{\frac{e^{m \cdot \log k} \cdot \frac{a}{k}}{k}}{k}\right) \cdot 10\right)\right)\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 3.333212644978590011114258413451524263306 \cdot 10^{107}:\\
\;\;\;\;\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\

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

\end{array}
double f(double a, double k, double m) {
        double r10315455 = a;
        double r10315456 = k;
        double r10315457 = m;
        double r10315458 = pow(r10315456, r10315457);
        double r10315459 = r10315455 * r10315458;
        double r10315460 = 1.0;
        double r10315461 = 10.0;
        double r10315462 = r10315461 * r10315456;
        double r10315463 = r10315460 + r10315462;
        double r10315464 = r10315456 * r10315456;
        double r10315465 = r10315463 + r10315464;
        double r10315466 = r10315459 / r10315465;
        return r10315466;
}
↓
double f(double a, double k, double m) {
        double r10315467 = k;
        double r10315468 = 3.33321264497859e+107;
        bool r10315469 = r10315467 <= r10315468;
        double r10315470 = m;
        double r10315471 = pow(r10315467, r10315470);
        double r10315472 = a;
        double r10315473 = r10315471 * r10315472;
        double r10315474 = 10.0;
        double r10315475 = r10315467 + r10315474;
        double r10315476 = 1.0;
        double r10315477 = fma(r10315467, r10315475, r10315476);
        double r10315478 = r10315473 / r10315477;
        double r10315479 = 99.0;
        double r10315480 = r10315467 * r10315467;
        double r10315481 = r10315479 / r10315480;
        double r10315482 = log(r10315467);
        double r10315483 = r10315470 * r10315482;
        double r10315484 = exp(r10315483);
        double r10315485 = r10315472 / r10315467;
        double r10315486 = r10315484 * r10315485;
        double r10315487 = r10315486 / r10315467;
        double r10315488 = sqrt(r10315467);
        double r10315489 = r10315485 / r10315488;
        double r10315490 = r10315484 / r10315488;
        double r10315491 = r10315487 / r10315467;
        double r10315492 = -r10315491;
        double r10315493 = r10315492 * r10315474;
        double r10315494 = fma(r10315489, r10315490, r10315493);
        double r10315495 = fma(r10315481, r10315487, r10315494);
        double r10315496 = r10315469 ? r10315478 : r10315495;
        return r10315496;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 3.33321264497859e+107

    1. Initial program 0.1

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

    2. Simplified0.1

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

    if 3.33321264497859e+107 < k

    1. Initial program 7.4

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

    2. Simplified7.4

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

    3. Taylor expanded around inf 7.4

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

    4. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{99}{k \cdot k}, \frac{\frac{a}{k} \cdot e^{\log k \cdot m}}{k}, \frac{\frac{a}{k} \cdot e^{\log k \cdot m}}{k} + \frac{\frac{\frac{a}{k} \cdot e^{\log k \cdot m}}{k}}{k} \cdot \left(-10\right)\right)}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt0.1

      \[\leadsto \mathsf{fma}\left(\frac{99}{k \cdot k}, \frac{\frac{a}{k} \cdot e^{\log k \cdot m}}{k}, \frac{\frac{a}{k} \cdot e^{\log k \cdot m}}{\color{blue}{\sqrt{k} \cdot \sqrt{k}}} + \frac{\frac{\frac{a}{k} \cdot e^{\log k \cdot m}}{k}}{k} \cdot \left(-10\right)\right)\]

    7. Applied times-frac0.2

      \[\leadsto \mathsf{fma}\left(\frac{99}{k \cdot k}, \frac{\frac{a}{k} \cdot e^{\log k \cdot m}}{k}, \color{blue}{\frac{\frac{a}{k}}{\sqrt{k}} \cdot \frac{e^{\log k \cdot m}}{\sqrt{k}}} + \frac{\frac{\frac{a}{k} \cdot e^{\log k \cdot m}}{k}}{k} \cdot \left(-10\right)\right)\]

    8. Applied fma-def0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 3.333212644978590011114258413451524263306 \cdot 10^{107}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{99}{k \cdot k}, \frac{e^{m \cdot \log k} \cdot \frac{a}{k}}{k}, \mathsf{fma}\left(\frac{\frac{a}{k}}{\sqrt{k}}, \frac{e^{m \cdot \log k}}{\sqrt{k}}, \left(-\frac{\frac{e^{m \cdot \log k} \cdot \frac{a}{k}}{k}}{k}\right) \cdot 10\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))