Average Error: 2.1 → 2.1
Time: 5.8s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{\left(\left(a \cdot {k}^{\left(\frac{m}{2}\right)}\right) \cdot \left(\sqrt[3]{{k}^{\left(\frac{m}{2}\right)}} \cdot \sqrt[3]{{k}^{\left(\frac{m}{2}\right)}}\right)\right) \cdot \sqrt[3]{{k}^{\left(\frac{m}{2}\right)}}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\frac{\left(\left(a \cdot {k}^{\left(\frac{m}{2}\right)}\right) \cdot \left(\sqrt[3]{{k}^{\left(\frac{m}{2}\right)}} \cdot \sqrt[3]{{k}^{\left(\frac{m}{2}\right)}}\right)\right) \cdot \sqrt[3]{{k}^{\left(\frac{m}{2}\right)}}}{\left(1 + 10 \cdot k\right) + k \cdot k}
double code(double a, double k, double m) {
	return ((double) (((double) (a * ((double) pow(k, m)))) / ((double) (((double) (1.0 + ((double) (10.0 * k)))) + ((double) (k * k))))));
}
↓
double code(double a, double k, double m) {
	return ((double) (((double) (((double) (((double) (a * ((double) pow(k, ((double) (m / 2.0)))))) * ((double) (((double) cbrt(((double) pow(k, ((double) (m / 2.0)))))) * ((double) cbrt(((double) pow(k, ((double) (m / 2.0)))))))))) * ((double) cbrt(((double) pow(k, ((double) (m / 2.0)))))))) / ((double) (((double) (1.0 + ((double) (10.0 * k)))) + ((double) (k * k))))));
}

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 2.1

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

  3. Applied sqr-pow2.1

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

  4. Applied associate-*r*2.1

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

  6. Applied add-cube-cbrt2.1

    \[\leadsto \frac{\left(a \cdot {k}^{\left(\frac{m}{2}\right)}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{{k}^{\left(\frac{m}{2}\right)}} \cdot \sqrt[3]{{k}^{\left(\frac{m}{2}\right)}}\right) \cdot \sqrt[3]{{k}^{\left(\frac{m}{2}\right)}}\right)}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]

  7. Applied associate-*r*2.1

    \[\leadsto \frac{\color{blue}{\left(\left(a \cdot {k}^{\left(\frac{m}{2}\right)}\right) \cdot \left(\sqrt[3]{{k}^{\left(\frac{m}{2}\right)}} \cdot \sqrt[3]{{k}^{\left(\frac{m}{2}\right)}}\right)\right) \cdot \sqrt[3]{{k}^{\left(\frac{m}{2}\right)}}}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]

  8. Final simplification2.1

    \[\leadsto \frac{\left(\left(a \cdot {k}^{\left(\frac{m}{2}\right)}\right) \cdot \left(\sqrt[3]{{k}^{\left(\frac{m}{2}\right)}} \cdot \sqrt[3]{{k}^{\left(\frac{m}{2}\right)}}\right)\right) \cdot \sqrt[3]{{k}^{\left(\frac{m}{2}\right)}}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]

Reproduce

herbie shell --seed 2020148 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))