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basic.lean
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basic.lean
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/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import data.rat
import data.semiquot
/-!
# Implementation of floating-point numbers (experimental).
-/
def int.shift2 (a b : ℕ) : ℤ → ℕ × ℕ
| (int.of_nat e) := (a.shiftl e, b)
| -[1+ e] := (a, b.shiftl e.succ)
namespace fp
@[derive inhabited]
inductive rmode
| NE -- round to nearest even
class float_cfg :=
(prec emax : ℕ)
(prec_pos : 0 < prec)
(prec_max : prec ≤ emax)
variable [C : float_cfg]
include C
def prec := C.prec
def emax := C.emax
def emin : ℤ := 1 - C.emax
def valid_finite (e : ℤ) (m : ℕ) : Prop :=
emin ≤ e + prec - 1 ∧ e + prec - 1 ≤ emax ∧ e = max (e + m.size - prec) emin
instance dec_valid_finite (e m) : decidable (valid_finite e m) :=
by unfold valid_finite; apply_instance
inductive float
| inf : bool → float
| nan : float
| finite : bool → Π e m, valid_finite e m → float
def float.is_finite : float → bool
| (float.finite s e m f) := tt
| _ := ff
def to_rat : Π (f : float), f.is_finite → ℚ
| (float.finite s e m f) _ :=
let (n, d) := int.shift2 m 1 e,
r := rat.mk_nat n d in
if s then -r else r
theorem float.zero.valid : valid_finite emin 0 :=
⟨begin
rw add_sub_assoc,
apply le_add_of_nonneg_right,
apply sub_nonneg_of_le,
apply int.coe_nat_le_coe_nat_of_le,
exact C.prec_pos
end,
suffices prec ≤ 2 * emax,
begin
rw ← int.coe_nat_le at this,
rw ← sub_nonneg at *,
simp only [emin, emax] at *,
ring_nf,
assumption
end, le_trans C.prec_max (nat.le_mul_of_pos_left dec_trivial),
by rw max_eq_right; simp [sub_eq_add_neg]⟩
def float.zero (s : bool) : float :=
float.finite s emin 0 float.zero.valid
instance : inhabited float := ⟨float.zero tt⟩
protected def float.sign' : float → semiquot bool
| (float.inf s) := pure s
| float.nan := ⊤
| (float.finite s e m f) := pure s
protected def float.sign : float → bool
| (float.inf s) := s
| float.nan := ff
| (float.finite s e m f) := s
protected def float.is_zero : float → bool
| (float.finite s e 0 f) := tt
| _ := ff
protected def float.neg : float → float
| (float.inf s) := float.inf (bnot s)
| float.nan := float.nan
| (float.finite s e m f) := float.finite (bnot s) e m f
def div_nat_lt_two_pow (n d : ℕ) : ℤ → bool
| (int.of_nat e) := n < d.shiftl e
| -[1+ e] := n.shiftl e.succ < d
-- TODO(Mario): Prove these and drop 'meta'
meta def of_pos_rat_dn (n : ℕ+) (d : ℕ+) : float × bool :=
begin
let e₁ : ℤ := n.1.size - d.1.size - prec,
cases h₁ : int.shift2 d.1 n.1 (e₁ + prec) with d₁ n₁,
let e₂ := if n₁ < d₁ then e₁ - 1 else e₁,
let e₃ := max e₂ emin,
cases h₂ : int.shift2 d.1 n.1 (e₃ + prec) with d₂ n₂,
let r := rat.mk_nat n₂ d₂,
let m := r.floor,
refine (float.finite ff e₃ (int.to_nat m) _, r.denom = 1),
{ exact undefined }
end
meta def next_up_pos (e m) (v : valid_finite e m) : float :=
let m' := m.succ in
if ss : m'.size = m.size then
float.finite ff e m' (by unfold valid_finite at *; rw ss; exact v)
else if h : e = emax then
float.inf ff
else
float.finite ff e.succ (nat.div2 m') undefined
meta def next_dn_pos (e m) (v : valid_finite e m) : float :=
match m with
| 0 := next_up_pos _ _ float.zero.valid
| nat.succ m' :=
if ss : m'.size = m.size then
float.finite ff e m' (by unfold valid_finite at *; rw ss; exact v)
else if h : e = emin then
float.finite ff emin m' undefined
else
float.finite ff e.pred (bit1 m') undefined
end
meta def next_up : float → float
| (float.finite ff e m f) := next_up_pos e m f
| (float.finite tt e m f) := float.neg $ next_dn_pos e m f
| f := f
meta def next_dn : float → float
| (float.finite ff e m f) := next_dn_pos e m f
| (float.finite tt e m f) := float.neg $ next_up_pos e m f
| f := f
meta def of_rat_up : ℚ → float
| ⟨0, _, _, _⟩ := float.zero ff
| ⟨nat.succ n, d, h, _⟩ :=
let (f, exact) := of_pos_rat_dn n.succ_pnat ⟨d, h⟩ in
if exact then f else next_up f
| ⟨-[1+n], d, h, _⟩ := float.neg (of_pos_rat_dn n.succ_pnat ⟨d, h⟩).1
meta def of_rat_dn (r : ℚ) : float :=
float.neg $ of_rat_up (-r)
meta def of_rat : rmode → ℚ → float
| rmode.NE r :=
let low := of_rat_dn r, high := of_rat_up r in
if hf : high.is_finite then
if r = to_rat _ hf then high else
if lf : low.is_finite then
if r - to_rat _ lf > to_rat _ hf - r then high else
if r - to_rat _ lf < to_rat _ hf - r then low else
match low, lf with float.finite s e m f, _ :=
if 2 ∣ m then low else high
end
else float.inf tt
else float.inf ff
namespace float
instance : has_neg float := ⟨float.neg⟩
meta def add (mode : rmode) : float → float → float
| nan _ := nan
| _ nan := nan
| (inf tt) (inf ff) := nan
| (inf ff) (inf tt) := nan
| (inf s₁) _ := inf s₁
| _ (inf s₂) := inf s₂
| (finite s₁ e₁ m₁ v₁) (finite s₂ e₂ m₂ v₂) :=
let f₁ := finite s₁ e₁ m₁ v₁, f₂ := finite s₂ e₂ m₂ v₂ in
of_rat mode (to_rat f₁ rfl + to_rat f₂ rfl)
meta instance : has_add float := ⟨float.add rmode.NE⟩
meta def sub (mode : rmode) (f1 f2 : float) : float :=
add mode f1 (-f2)
meta instance : has_sub float := ⟨float.sub rmode.NE⟩
meta def mul (mode : rmode) : float → float → float
| nan _ := nan
| _ nan := nan
| (inf s₁) f₂ := if f₂.is_zero then nan else inf (bxor s₁ f₂.sign)
| f₁ (inf s₂) := if f₁.is_zero then nan else inf (bxor f₁.sign s₂)
| (finite s₁ e₁ m₁ v₁) (finite s₂ e₂ m₂ v₂) :=
let f₁ := finite s₁ e₁ m₁ v₁, f₂ := finite s₂ e₂ m₂ v₂ in
of_rat mode (to_rat f₁ rfl * to_rat f₂ rfl)
meta def div (mode : rmode) : float → float → float
| nan _ := nan
| _ nan := nan
| (inf s₁) (inf s₂) := nan
| (inf s₁) f₂ := inf (bxor s₁ f₂.sign)
| f₁ (inf s₂) := zero (bxor f₁.sign s₂)
| (finite s₁ e₁ m₁ v₁) (finite s₂ e₂ m₂ v₂) :=
let f₁ := finite s₁ e₁ m₁ v₁, f₂ := finite s₂ e₂ m₂ v₂ in
if f₂.is_zero then inf (bxor s₁ s₂) else
of_rat mode (to_rat f₁ rfl / to_rat f₂ rfl)
end float
end fp