Nicer printing for decimal constants in Q

Print 1.5 as 1.5 and not 15e-1.

We choose the shortest representation
with tie break to the dot notation (0.01 rather than 1e-3).

The printing remains injective, i.e. 12/10 is not mixed with 120/100,
the first being printed as 1.2 and the last as 1.20.

3 files changed, 68 insertions, 11 deletions

diff --git a/test-suite/output/QArithSyntax.out b/test-suite/output/QArithSyntax.out
index 6bc04f1cef..fe6a1d25c6 100644
--- a/test-suite/output/QArithSyntax.out
+++ b/test-suite/output/QArithSyntax.out
@@ -1,14 +1,14 @@
-eq_refl : 102e-2 = 102e-2
-     : 102e-2 = 102e-2
-eq_refl : 102e-1 = 102e-1
-     : 102e-1 = 102e-1
+eq_refl : 1.02 = 1.02
+     : 1.02 = 1.02
+eq_refl : 10.2 = 10.2
+     : 10.2 = 10.2
 eq_refl : 1020 = 1020
      : 1020 = 1020
 eq_refl : 102 = 102
      : 102 = 102
-eq_refl : 102e-2 = 102e-2
-     : 102e-2 = 102e-2
+eq_refl : 1.02 = 1.02
+     : 1.02 = 1.02
 eq_refl : -1e-4 = -1e-4
      : -1e-4 = -1e-4
-eq_refl : -50e-2 = -50e-2
-     : -50e-2 = -50e-2
+eq_refl : -0.50 = -0.50
+     : -0.50 = -0.50
diff --git a/theories/Init/Decimal.v b/theories/Init/Decimal.v
index 10c3baa2cd..855db8bc3f 100644
--- a/theories/Init/Decimal.v
+++ b/theories/Init/Decimal.v
@@ -156,6 +156,37 @@ Definition nztail_int d :=
   | Neg d => let (r, n) := nztail d in pair (Neg r) n
   end.
 
+(** [del_head n d] removes [n] digits at beginning of [d]
+    or returns [zero] if [d] has less than [n] digits. *)
+
+Fixpoint del_head n d :=
+  match n with
+  | O => d
+  | S n =>
+    match d with
+    | Nil => zero
+    | D0 d | D1 d | D2 d | D3 d | D4 d | D5 d | D6 d | D7 d | D8 d | D9 d =>
+      del_head n d
+    end
+  end.
+
+Definition del_head_int n d :=
+  match d with
+  | Pos d => Pos (del_head n d)
+  | Neg d => Neg (del_head n d)
+  end.
+
+(** [del_tail n d] removes [n] digits at end of [d]
+    or returns [zero] if [d] has less than [n] digits. *)
+
+Fixpoint del_tail n d := rev (del_head n (rev d)).
+
+Definition del_tail_int n d :=
+  match d with
+  | Pos d => Pos (del_tail n d)
+  | Neg d => Neg (del_tail n d)
+  end.
+
 Module Little.
 
 (** Successor of little-endian numbers *)
diff --git a/theories/QArith/QArith_base.v b/theories/QArith/QArith_base.v
index a7f338aec3..bd5225d9ef 100644
--- a/theories/QArith/QArith_base.v
+++ b/theories/QArith/QArith_base.v
@@ -44,13 +44,39 @@ Definition of_decimal (d:Decimal.decimal) : Q :=
   end.
 
 Definition to_decimal (q:Q) : option Decimal.decimal :=
+  (* choose between 123e-2 and 1.23, this is purely heuristic
+     and doesn't play any soundness role *)
+  let choose_exponent i ne :=
+    let i := match i with Decimal.Pos i | Decimal.Neg i => i end in
+    let li := Decimal.nb_digits i in
+    let le := Decimal.nb_digits (Nat.to_uint ne) in
+    Nat.ltb (Nat.add li le) ne in
+  (* print 123 / 100 as 123e-2 *)
+  let decimal_exponent i ne :=
+      let e := Z.to_int (Z.opp (Z.of_nat ne)) in
+      Decimal.DecimalExp i Decimal.Nil e in
+  (* print 123 / 100 as 1.23 *)
+  let decimal_dot i ne :=
+    let ai := match i with Decimal.Pos i | Decimal.Neg i => i end in
+    let ni := Decimal.nb_digits ai in
+    if Nat.ltb ne ni then
+      let i := Decimal.del_tail_int ne i in
+      let f := Decimal.del_head (Nat.sub ni ne) ai in
+      Decimal.Decimal i f
+    else
+      let z := match i with
+        | Decimal.Pos _ => Decimal.Pos (Decimal.zero)
+        | Decimal.Neg _ => Decimal.Neg (Decimal.zero) end in
+      Decimal.Decimal z (Nat.iter (Nat.sub ne ni) Decimal.D0 ai) in
   let num := Z.to_int (Qnum q) in
   let (den, e_den) := Decimal.nztail (Pos.to_uint (Qden q)) in
   match den with
   | Decimal.D1 Decimal.Nil =>
-    match Z.of_nat e_den with
-    | Z0 => Some (Decimal.Decimal num Decimal.Nil)
-    | e => Some (Decimal.DecimalExp num Decimal.Nil (Z.to_int (Z.opp e)))
+    match e_den with
+    | O => Some (Decimal.Decimal num Decimal.Nil)
+    | ne =>
+      if choose_exponent num ne then Some (decimal_exponent num ne)
+      else Some (decimal_dot num ne)
     end
   | _ => None
   end.