Definition binary_float_of_bits_aux x :=
let '(sx, mx, ex) := split_bits x in
if Zeq_bool ex 0 then
match mx with
| Z0 => F754_zero sx
| Zpos px => F754_finite sx px emin
| Zneg _ => F754_nan false xH (* dummy *)
end
else if Zeq_bool ex (Zpower 2 ew - 1) then
match mx with
| Z0 => F754_infinity sx
| Zpos plx => F754_nan sx plx
| Zneg _ => F754_nan false xH (* dummy *)
end
else
match (mx + Zpower 2 mw)%Z with
| Zpos px => F754_finite sx px (ex + emin - 1)
| _ => F754_nan false xH (* dummy *)
end.

Lemma binary_float_of_bits_aux_correct :
forall x,
valid_binary prec emax (binary_float_of_bits_aux x) = true.