Average Error: 1.8 → 1.9
Time: 4.8s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{\left(a \cdot {k}^{\left(\frac{m}{2}\right)}\right) \cdot {k}^{\left(\frac{m}{2}\right)}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot \sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}}\]
\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)}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot \sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}}
double f(double a, double k, double m) {
        double r210725 = a;
        double r210726 = k;
        double r210727 = m;
        double r210728 = pow(r210726, r210727);
        double r210729 = r210725 * r210728;
        double r210730 = 1.0;
        double r210731 = 10.0;
        double r210732 = r210731 * r210726;
        double r210733 = r210730 + r210732;
        double r210734 = r210726 * r210726;
        double r210735 = r210733 + r210734;
        double r210736 = r210729 / r210735;
        return r210736;
}
↓
double f(double a, double k, double m) {
        double r210737 = a;
        double r210738 = k;
        double r210739 = m;
        double r210740 = 2.0;
        double r210741 = r210739 / r210740;
        double r210742 = pow(r210738, r210741);
        double r210743 = r210737 * r210742;
        double r210744 = r210743 * r210742;
        double r210745 = 1.0;
        double r210746 = 10.0;
        double r210747 = r210746 * r210738;
        double r210748 = r210745 + r210747;
        double r210749 = r210738 * r210738;
        double r210750 = r210748 + r210749;
        double r210751 = sqrt(r210750);
        double r210752 = r210751 * r210751;
        double r210753 = r210744 / r210752;
        return r210753;
}

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 1.8

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

  3. Applied sqr-pow1.8

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

  4. Applied associate-*r*1.8

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

  6. Applied add-sqr-sqrt1.9

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

  7. Final simplification1.9

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

Reproduce

herbie shell --seed 2020100 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))