Average Error: 2.3 → 2.4
Time: 4.9s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\left(a \cdot {k}^{m}\right) \cdot {\left(\sqrt{1 + k \cdot \left(10 + k\right)}\right)}^{\left(-2\right)}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\left(a \cdot {k}^{m}\right) \cdot {\left(\sqrt{1 + k \cdot \left(10 + k\right)}\right)}^{\left(-2\right)}
double code(double a, double k, double m) {
	return ((a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k)));
}
↓
double code(double a, double k, double m) {
	return ((a * pow(k, m)) * pow(sqrt((1.0 + (k * (10.0 + k)))), -2.0));
}

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 2.3

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

  3. Applied div-inv2.4

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

  4. Simplified2.3

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

  6. Applied add-sqr-sqrt2.4

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

  7. Applied associate-/r*2.4

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

  9. Applied pow12.4

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

  10. Applied pow12.4

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

  11. Applied pow-flip2.4

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

  12. Applied pow-div2.4

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

  13. Simplified2.4

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

  14. Final simplification2.4

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

Reproduce

herbie shell --seed 2020057 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))