Average Error: 2.0 → 2.0
Time: 52.6s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[a \cdot \frac{1}{\frac{\mathsf{fma}\left(k, 10 + k, 1\right)}{{k}^{m}}}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
a \cdot \frac{1}{\frac{\mathsf{fma}\left(k, 10 + k, 1\right)}{{k}^{m}}}
double f(double a, double k, double m) {
        double r17510554 = a;
        double r17510555 = k;
        double r17510556 = m;
        double r17510557 = pow(r17510555, r17510556);
        double r17510558 = r17510554 * r17510557;
        double r17510559 = 1.0;
        double r17510560 = 10.0;
        double r17510561 = r17510560 * r17510555;
        double r17510562 = r17510559 + r17510561;
        double r17510563 = r17510555 * r17510555;
        double r17510564 = r17510562 + r17510563;
        double r17510565 = r17510558 / r17510564;
        return r17510565;
}
↓
double f(double a, double k, double m) {
        double r17510566 = a;
        double r17510567 = 1.0;
        double r17510568 = k;
        double r17510569 = 10.0;
        double r17510570 = r17510569 + r17510568;
        double r17510571 = 1.0;
        double r17510572 = fma(r17510568, r17510570, r17510571);
        double r17510573 = m;
        double r17510574 = pow(r17510568, r17510573);
        double r17510575 = r17510572 / r17510574;
        double r17510576 = r17510567 / r17510575;
        double r17510577 = r17510566 * r17510576;
        return r17510577;
}

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

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

  4. Applied div-inv2.0

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

  5. Final simplification2.0

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))