Average Error: 2.0 → 0.1
Time: 24.1s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 65498774778259760:\\ \;\;\;\;\frac{\left({\left(\sqrt[3]{k}\right)}^{m} \cdot a\right) \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{a}{k}}{k} \cdot e^{m \cdot \log k}\right) \cdot \left(\frac{\frac{99}{k}}{k} - \frac{10}{k}\right) + \frac{\frac{a}{k}}{k} \cdot e^{m \cdot \log k}\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 65498774778259760:\\
\;\;\;\;\frac{\left({\left(\sqrt[3]{k}\right)}^{m} \cdot a\right) \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\

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

\end{array}
double f(double a, double k, double m) {
        double r8102349 = a;
        double r8102350 = k;
        double r8102351 = m;
        double r8102352 = pow(r8102350, r8102351);
        double r8102353 = r8102349 * r8102352;
        double r8102354 = 1.0;
        double r8102355 = 10.0;
        double r8102356 = r8102355 * r8102350;
        double r8102357 = r8102354 + r8102356;
        double r8102358 = r8102350 * r8102350;
        double r8102359 = r8102357 + r8102358;
        double r8102360 = r8102353 / r8102359;
        return r8102360;
}
↓
double f(double a, double k, double m) {
        double r8102361 = k;
        double r8102362 = 6.549877477825976e+16;
        bool r8102363 = r8102361 <= r8102362;
        double r8102364 = cbrt(r8102361);
        double r8102365 = m;
        double r8102366 = pow(r8102364, r8102365);
        double r8102367 = a;
        double r8102368 = r8102366 * r8102367;
        double r8102369 = r8102364 * r8102364;
        double r8102370 = pow(r8102369, r8102365);
        double r8102371 = r8102368 * r8102370;
        double r8102372 = 10.0;
        double r8102373 = r8102361 + r8102372;
        double r8102374 = 1.0;
        double r8102375 = fma(r8102361, r8102373, r8102374);
        double r8102376 = r8102371 / r8102375;
        double r8102377 = r8102367 / r8102361;
        double r8102378 = r8102377 / r8102361;
        double r8102379 = log(r8102361);
        double r8102380 = r8102365 * r8102379;
        double r8102381 = exp(r8102380);
        double r8102382 = r8102378 * r8102381;
        double r8102383 = 99.0;
        double r8102384 = r8102383 / r8102361;
        double r8102385 = r8102384 / r8102361;
        double r8102386 = r8102372 / r8102361;
        double r8102387 = r8102385 - r8102386;
        double r8102388 = r8102382 * r8102387;
        double r8102389 = r8102388 + r8102382;
        double r8102390 = r8102363 ? r8102376 : r8102389;
        return r8102390;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 6.549877477825976e+16

    1. Initial program 0.1

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

    2. Simplified0.0

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

    4. Applied add-cube-cbrt0.0

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

    5. Applied unpow-prod-down0.0

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

    6. Applied associate-*l*0.1

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

    if 6.549877477825976e+16 < k

    1. Initial program 5.2

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

    2. Simplified5.2

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

    3. Taylor expanded around -inf 64.0

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

    4. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 65498774778259760:\\ \;\;\;\;\frac{\left({\left(\sqrt[3]{k}\right)}^{m} \cdot a\right) \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{a}{k}}{k} \cdot e^{m \cdot \log k}\right) \cdot \left(\frac{\frac{99}{k}}{k} - \frac{10}{k}\right) + \frac{\frac{a}{k}}{k} \cdot e^{m \cdot \log k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))