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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `a`

Derivation

Initial program 2.0
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
Using strategy rm
Applied add-sqr-sqrt2.1
\[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot \sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}}}\]
Applied associate-/r*2.1
\[\leadsto \color{blue}{\frac{\frac{a \cdot {k}^{m}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}}}\]
Applied simplify2.1
\[\leadsto \frac{\color{blue}{\frac{{k}^{m} \cdot a}{\sqrt{(k \cdot \left(10 + k\right) + 1)_*}}}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}}\]

Runtime

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))