Average Error: 2.1 → 2.0
Time: 16.2s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{a}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{{k}^{m}}}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\frac{a}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{{k}^{m}}}
double f(double a, double k, double m) {
        double r9434042 = a;
        double r9434043 = k;
        double r9434044 = m;
        double r9434045 = pow(r9434043, r9434044);
        double r9434046 = r9434042 * r9434045;
        double r9434047 = 1.0;
        double r9434048 = 10.0;
        double r9434049 = r9434048 * r9434043;
        double r9434050 = r9434047 + r9434049;
        double r9434051 = r9434043 * r9434043;
        double r9434052 = r9434050 + r9434051;
        double r9434053 = r9434046 / r9434052;
        return r9434053;
}
↓
double f(double a, double k, double m) {
        double r9434054 = a;
        double r9434055 = k;
        double r9434056 = 10.0;
        double r9434057 = r9434055 + r9434056;
        double r9434058 = 1.0;
        double r9434059 = fma(r9434055, r9434057, r9434058);
        double r9434060 = m;
        double r9434061 = pow(r9434055, r9434060);
        double r9434062 = r9434059 / r9434061;
        double r9434063 = r9434054 / r9434062;
        return r9434063;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Initial program 2.1

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

  2. Simplified2.0

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

  3. Final simplification2.0

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))