\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

↓

\[w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}}\]

Error

The X axis uses an exponential scale — Bits error versus `w0`

Derivation

Initial program 14.1
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
Simplified12.4
\[\leadsto \color{blue}{\sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}} \cdot w0}\]
Using strategy rm
Applied associate-*l/8.3
\[\leadsto \sqrt{1 - \color{blue}{\frac{h \cdot \frac{M \cdot D}{2 \cdot d}}{\ell}} \cdot \frac{M \cdot D}{2 \cdot d}} \cdot w0\]
Final simplification8.3
\[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}}\]

Reproduce

herbie shell --seed 2019090 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))