Average Error: 2.0 → 2.0
Time: 1.4m
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 r59497075 = a;
        double r59497076 = k;
        double r59497077 = m;
        double r59497078 = pow(r59497076, r59497077);
        double r59497079 = r59497075 * r59497078;
        double r59497080 = 1.0;
        double r59497081 = 10.0;
        double r59497082 = r59497081 * r59497076;
        double r59497083 = r59497080 + r59497082;
        double r59497084 = r59497076 * r59497076;
        double r59497085 = r59497083 + r59497084;
        double r59497086 = r59497079 / r59497085;
        return r59497086;
}
↓
double f(double a, double k, double m) {
        double r59497087 = k;
        double r59497088 = m;
        double r59497089 = pow(r59497087, r59497088);
        double r59497090 = a;
        double r59497091 = r59497089 * r59497090;
        double r59497092 = 10.0;
        double r59497093 = r59497092 + r59497087;
        double r59497094 = 1.0;
        double r59497095 = fma(r59497093, r59497087, r59497094);
        double r59497096 = r59497091 / r59497095;
        return r59497096;
}

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))))