Average Error: 2.0 → 2.0
Time: 14.9s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(\left(10 + k\right), k, 1\right)}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(\left(10 + k\right), k, 1\right)}
double f(double a, double k, double m) {
        double r2892790 = a;
        double r2892791 = k;
        double r2892792 = m;
        double r2892793 = pow(r2892791, r2892792);
        double r2892794 = r2892790 * r2892793;
        double r2892795 = 1.0;
        double r2892796 = 10.0;
        double r2892797 = r2892796 * r2892791;
        double r2892798 = r2892795 + r2892797;
        double r2892799 = r2892791 * r2892791;
        double r2892800 = r2892798 + r2892799;
        double r2892801 = r2892794 / r2892800;
        return r2892801;
}
↓
double f(double a, double k, double m) {
        double r2892802 = k;
        double r2892803 = m;
        double r2892804 = pow(r2892802, r2892803);
        double r2892805 = a;
        double r2892806 = r2892804 * r2892805;
        double r2892807 = 10.0;
        double r2892808 = r2892807 + r2892802;
        double r2892809 = 1.0;
        double r2892810 = fma(r2892808, r2892802, r2892809);
        double r2892811 = r2892806 / r2892810;
        return r2892811;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Initial program 2.0

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

  2. Simplified2.0

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

  3. Final simplification2.0

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))