Average Error: 2.3 → 2.3
Time: 31.2s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{{\left(\sqrt[3]{k}\right)}^{m} \cdot \left({\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m} \cdot a\right)}{\mathsf{fma}\left(k, k + 10, 1\right)}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\frac{{\left(\sqrt[3]{k}\right)}^{m} \cdot \left({\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m} \cdot a\right)}{\mathsf{fma}\left(k, k + 10, 1\right)}
double f(double a, double k, double m) {
        double r5926689 = a;
        double r5926690 = k;
        double r5926691 = m;
        double r5926692 = pow(r5926690, r5926691);
        double r5926693 = r5926689 * r5926692;
        double r5926694 = 1.0;
        double r5926695 = 10.0;
        double r5926696 = r5926695 * r5926690;
        double r5926697 = r5926694 + r5926696;
        double r5926698 = r5926690 * r5926690;
        double r5926699 = r5926697 + r5926698;
        double r5926700 = r5926693 / r5926699;
        return r5926700;
}
↓
double f(double a, double k, double m) {
        double r5926701 = k;
        double r5926702 = cbrt(r5926701);
        double r5926703 = m;
        double r5926704 = pow(r5926702, r5926703);
        double r5926705 = r5926702 * r5926702;
        double r5926706 = pow(r5926705, r5926703);
        double r5926707 = a;
        double r5926708 = r5926706 * r5926707;
        double r5926709 = r5926704 * r5926708;
        double r5926710 = 10.0;
        double r5926711 = r5926701 + r5926710;
        double r5926712 = 1.0;
        double r5926713 = fma(r5926701, r5926711, r5926712);
        double r5926714 = r5926709 / r5926713;
        return r5926714;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Initial program 2.3

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

  2. Simplified2.3

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

  4. Applied add-cube-cbrt2.3

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

  5. Applied unpow-prod-down2.3

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

  6. Applied associate-*l*2.3

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

  8. Applied *-un-lft-identity2.3

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

  9. Applied associate-/r*2.3

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

  10. Simplified2.3

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

  11. Final simplification2.3

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))