Average Error: 2.2 → 2.2
Time: 4.4s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{\left(a \cdot {k}^{\left(\frac{m}{2}\right)}\right) \cdot {k}^{\left(\frac{m}{2}\right)}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\frac{\left(a \cdot {k}^{\left(\frac{m}{2}\right)}\right) \cdot {k}^{\left(\frac{m}{2}\right)}}{\left(1 + 10 \cdot k\right) + k \cdot k}
double f(double a, double k, double m) {
        double r161542 = a;
        double r161543 = k;
        double r161544 = m;
        double r161545 = pow(r161543, r161544);
        double r161546 = r161542 * r161545;
        double r161547 = 1.0;
        double r161548 = 10.0;
        double r161549 = r161548 * r161543;
        double r161550 = r161547 + r161549;
        double r161551 = r161543 * r161543;
        double r161552 = r161550 + r161551;
        double r161553 = r161546 / r161552;
        return r161553;
}
↓
double f(double a, double k, double m) {
        double r161554 = a;
        double r161555 = k;
        double r161556 = m;
        double r161557 = 2.0;
        double r161558 = r161556 / r161557;
        double r161559 = pow(r161555, r161558);
        double r161560 = r161554 * r161559;
        double r161561 = r161560 * r161559;
        double r161562 = 1.0;
        double r161563 = 10.0;
        double r161564 = r161563 * r161555;
        double r161565 = r161562 + r161564;
        double r161566 = r161555 * r161555;
        double r161567 = r161565 + r161566;
        double r161568 = r161561 / r161567;
        return r161568;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 2.2

    \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
  2. Using strategy rm

  3. Applied sqr-pow2.2

    \[\leadsto \frac{a \cdot \color{blue}{\left({k}^{\left(\frac{m}{2}\right)} \cdot {k}^{\left(\frac{m}{2}\right)}\right)}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]

  4. Applied associate-*r*2.2

    \[\leadsto \frac{\color{blue}{\left(a \cdot {k}^{\left(\frac{m}{2}\right)}\right) \cdot {k}^{\left(\frac{m}{2}\right)}}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]

  5. Final simplification2.2

    \[\leadsto \frac{\left(a \cdot {k}^{\left(\frac{m}{2}\right)}\right) \cdot {k}^{\left(\frac{m}{2}\right)}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))