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

↓

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

Error

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

Derivation

Initial program 2.1
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
Applied simplify2.0
\[\leadsto \color{blue}{\frac{{k}^{m} \cdot a}{(\left(10 + k\right) \cdot k + 1)_*}}\]
Using strategy rm
Applied add-sqr-sqrt2.1
\[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{\sqrt{(\left(10 + k\right) \cdot k + 1)_*} \cdot \sqrt{(\left(10 + k\right) \cdot k + 1)_*}}}\]
Applied times-frac2.1
\[\leadsto \color{blue}{\frac{{k}^{m}}{\sqrt{(\left(10 + k\right) \cdot k + 1)_*}} \cdot \frac{a}{\sqrt{(\left(10 + k\right) \cdot k + 1)_*}}}\]

Runtime

herbie shell --seed 2018170 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))