Average Error: 2.1 → 2.1
Time: 36.5s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[{k}^{m} \cdot \frac{a}{\mathsf{fma}\left(\left(k + 10\right), k, 1\right)}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
{k}^{m} \cdot \frac{a}{\mathsf{fma}\left(\left(k + 10\right), k, 1\right)}
double f(double a, double k, double m) {
        double r65169415 = a;
        double r65169416 = k;
        double r65169417 = m;
        double r65169418 = pow(r65169416, r65169417);
        double r65169419 = r65169415 * r65169418;
        double r65169420 = 1.0;
        double r65169421 = 10.0;
        double r65169422 = r65169421 * r65169416;
        double r65169423 = r65169420 + r65169422;
        double r65169424 = r65169416 * r65169416;
        double r65169425 = r65169423 + r65169424;
        double r65169426 = r65169419 / r65169425;
        return r65169426;
}
↓
double f(double a, double k, double m) {
        double r65169427 = k;
        double r65169428 = m;
        double r65169429 = pow(r65169427, r65169428);
        double r65169430 = a;
        double r65169431 = 10.0;
        double r65169432 = r65169427 + r65169431;
        double r65169433 = 1.0;
        double r65169434 = fma(r65169432, r65169427, r65169433);
        double r65169435 = r65169430 / r65169434;
        double r65169436 = r65169429 * r65169435;
        return r65169436;
}

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. Simplified2.1

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

  4. Applied *-un-lft-identity2.1

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

  5. Applied times-frac2.1

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

  6. Simplified2.1

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

  7. Final simplification2.1

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))