Definition neg_pl (s:bool) (pl:nan_pl 24) := (negb s, pl).Definition abs_pl (s:bool) (pl:nan_pl 24) := (false, pl).Definition binop_pl (x y: binary32) : bool*nan_pl 24 :=
  match x, y with
  | B754_nan s1 pl1, B754_nan s2 pl2 =>
      if Archi.choose_binop_pl_32 s1 pl1 s2 pl2
      then (s2, transform_quiet_pl pl2)
      else (s1, transform_quiet_pl pl1)
  | B754_nan s1 pl1, _ => (s1, transform_quiet_pl pl1)
  | _, B754_nan s2 pl2 => (s2, transform_quiet_pl pl2)
  | _, _ => Archi.default_pl_32
  end.(** ** Operations over single-precision floats *)

Definition zero: float32 := B754_zero _ _ false.(**r the float [+0.0] *)

Definition eq_dec: forall (f1 f2: float32), {f1 = f2} + {f1 <> f2} := Beq_dec _ _.(** Arithmetic operations *)

Definition neg: float32 -> float32 := Bopp _ _ neg_pl.(**r opposite (change sign) *)
Definition abs: float32 -> float32 := Babs _ _ abs_pl.(**r absolute value (set sign to [+]) *)
Definition add: float32 -> float32 -> float32 :=
  Bplus 24 128 __ __ binop_pl mode_NE.(**r addition *)
Definition sub: float32 -> float32 -> float32 :=
  Bminus 24 128 __ __ binop_pl mode_NE.(**r subtraction *)
Definition mul: float32 -> float32 -> float32 :=
  Bmult 24 128 __ __ binop_pl mode_NE.(**r multiplication *)
Definition div: float32 -> float32 -> float32 :=
  Bdiv 24 128 __ __ binop_pl mode_NE.(**r division *)
Definition cmp (c:comparison) (f1 f2: float32) : bool := (**r comparison *)
  cmp_of_comparison c (Bcompare _ _ f1 f2).(** Conversions *)

Definition of_double : float -> float32 := Float.to_single.Definition to_double : float32 -> float := Float.of_single.Definition to_int (f:float32): option int := (**r conversion to signed 32-bit int *)
  option_map Int.repr (ZofB_range _ _ f Int.min_signed Int.max_signed).Definition to_intu (f:float32): option int := (**r conversion to unsigned 32-bit int *)
  option_map Int.repr (ZofB_range _ _ f 0 Int.max_unsigned).Definition to_long (f:float32): option int64 := (**r conversion to signed 64-bit int *)
  option_map Int64.repr (ZofB_range _ _ f Int64.min_signed Int64.max_signed).Definition to_longu (f:float32): option int64 := (**r conversion to unsigned 64-bit int *)
  option_map Int64.repr (ZofB_range _ _ f 0 Int64.max_unsigned).Definition of_int (n:int): float32 := (**r conversion from signed 32-bit int to single-precision float *)
  BofZ 24 128 __ __ (Int.signed n).Definition of_intu (n:int): float32 := (**r conversion from unsigned 32-bit int to single-precision float *)
  BofZ 24 128 __ __ (Int.unsigned n).Definition of_long (n:int64): float32 := (**r conversion from signed 64-bit int to single-precision float *)
  BofZ 24 128 __ __ (Int64.signed n).Definition of_longu (n:int64): float32 := (**r conversion from unsigned 64-bit int to single-precision float *)
  BofZ 24 128 __ __ (Int64.unsigned n).Definition from_parsed (base:positive) (intPart:positive) (expPart:Z) : float32 :=
  Bparse 24 128 __ __ base intPart expPart.(** Conversions between floats and their concrete in-memory representation
    as a sequence of 32 bits. *)

Definition to_bits (f: float32) : int := Int.repr (bits_of_b32 f).Definition of_bits (b: int): float32 := b32_of_bits (Int.unsigned b).

Theorem mul_commut: forall x y, is_nan _ _ x = false \/ is_nan _ _ y = false -> mul x y = mul y x.