606 lines
18 KiB
Ada
606 lines
18 KiB
Ada
------------------------------------------------------------------------------
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-- --
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-- GNAT RUN-TIME COMPONENTS --
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-- --
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-- S Y S T E M . A R I T H _ 6 4 --
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-- --
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-- B o d y --
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-- --
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-- Copyright (C) 1992-2015, Free Software Foundation, Inc. --
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-- --
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-- GNAT is free software; you can redistribute it and/or modify it under --
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-- terms of the GNU General Public License as published by the Free Soft- --
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-- ware Foundation; either version 3, or (at your option) any later ver- --
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-- sion. GNAT is distributed in the hope that it will be useful, but WITH- --
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-- OUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY --
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-- or FITNESS FOR A PARTICULAR PURPOSE. --
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-- --
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-- As a special exception under Section 7 of GPL version 3, you are granted --
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-- additional permissions described in the GCC Runtime Library Exception, --
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-- version 3.1, as published by the Free Software Foundation. --
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-- --
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-- You should have received a copy of the GNU General Public License and --
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-- a copy of the GCC Runtime Library Exception along with this program; --
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-- see the files COPYING3 and COPYING.RUNTIME respectively. If not, see --
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-- <http://www.gnu.org/licenses/>. --
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-- --
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-- GNAT was originally developed by the GNAT team at New York University. --
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-- Extensive contributions were provided by Ada Core Technologies Inc. --
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-- --
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------------------------------------------------------------------------------
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with Interfaces; use Interfaces;
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with Ada.Unchecked_Conversion;
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package body System.Arith_64 is
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pragma Suppress (Overflow_Check);
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pragma Suppress (Range_Check);
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subtype Uns64 is Unsigned_64;
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function To_Uns is new Ada.Unchecked_Conversion (Int64, Uns64);
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function To_Int is new Ada.Unchecked_Conversion (Uns64, Int64);
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subtype Uns32 is Unsigned_32;
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-----------------------
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-- Local Subprograms --
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-----------------------
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function "+" (A, B : Uns32) return Uns64 is (Uns64 (A) + Uns64 (B));
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function "+" (A : Uns64; B : Uns32) return Uns64 is (A + Uns64 (B));
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-- Length doubling additions
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function "*" (A, B : Uns32) return Uns64 is (Uns64 (A) * Uns64 (B));
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-- Length doubling multiplication
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function "/" (A : Uns64; B : Uns32) return Uns64 is (A / Uns64 (B));
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-- Length doubling division
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function "&" (Hi, Lo : Uns32) return Uns64 is
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(Shift_Left (Uns64 (Hi), 32) or Uns64 (Lo));
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-- Concatenate hi, lo values to form 64-bit result
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function "abs" (X : Int64) return Uns64 is
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(if X = Int64'First then 2**63 else Uns64 (Int64'(abs X)));
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-- Convert absolute value of X to unsigned. Note that we can't just use
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-- the expression of the Else, because it overflows for X = Int64'First.
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function "rem" (A : Uns64; B : Uns32) return Uns64 is (A rem Uns64 (B));
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-- Length doubling remainder
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function Le3 (X1, X2, X3 : Uns32; Y1, Y2, Y3 : Uns32) return Boolean;
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-- Determines if 96 bit value X1&X2&X3 <= Y1&Y2&Y3
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function Lo (A : Uns64) return Uns32 is (Uns32 (A and 16#FFFF_FFFF#));
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-- Low order half of 64-bit value
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function Hi (A : Uns64) return Uns32 is (Uns32 (Shift_Right (A, 32)));
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-- High order half of 64-bit value
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procedure Sub3 (X1, X2, X3 : in out Uns32; Y1, Y2, Y3 : Uns32);
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-- Computes X1&X2&X3 := X1&X2&X3 - Y1&Y1&Y3 with mod 2**96 wrap
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function To_Neg_Int (A : Uns64) return Int64 with Inline;
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-- Convert to negative integer equivalent. If the input is in the range
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-- 0 .. 2 ** 63, then the corresponding negative signed integer (obtained
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-- by negating the given value) is returned, otherwise constraint error
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-- is raised.
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function To_Pos_Int (A : Uns64) return Int64 with Inline;
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-- Convert to positive integer equivalent. If the input is in the range
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-- 0 .. 2 ** 63-1, then the corresponding non-negative signed integer is
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-- returned, otherwise constraint error is raised.
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procedure Raise_Error with Inline;
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pragma No_Return (Raise_Error);
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-- Raise constraint error with appropriate message
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--------------------------
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-- Add_With_Ovflo_Check --
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--------------------------
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function Add_With_Ovflo_Check (X, Y : Int64) return Int64 is
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R : constant Int64 := To_Int (To_Uns (X) + To_Uns (Y));
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begin
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if X >= 0 then
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if Y < 0 or else R >= 0 then
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return R;
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end if;
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else -- X < 0
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if Y > 0 or else R < 0 then
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return R;
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end if;
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end if;
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Raise_Error;
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end Add_With_Ovflo_Check;
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-------------------
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-- Double_Divide --
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-------------------
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procedure Double_Divide
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(X, Y, Z : Int64;
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Q, R : out Int64;
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Round : Boolean)
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is
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Xu : constant Uns64 := abs X;
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Yu : constant Uns64 := abs Y;
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Yhi : constant Uns32 := Hi (Yu);
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Ylo : constant Uns32 := Lo (Yu);
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Zu : constant Uns64 := abs Z;
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Zhi : constant Uns32 := Hi (Zu);
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Zlo : constant Uns32 := Lo (Zu);
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T1, T2 : Uns64;
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Du, Qu, Ru : Uns64;
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Den_Pos : Boolean;
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begin
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if Yu = 0 or else Zu = 0 then
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Raise_Error;
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end if;
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-- Compute Y * Z. Note that if the result overflows 64 bits unsigned,
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-- then the rounded result is clearly zero (since the dividend is at
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-- most 2**63 - 1, the extra bit of precision is nice here).
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if Yhi /= 0 then
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if Zhi /= 0 then
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Q := 0;
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R := X;
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return;
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else
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T2 := Yhi * Zlo;
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end if;
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else
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T2 := (if Zhi /= 0 then Ylo * Zhi else 0);
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end if;
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T1 := Ylo * Zlo;
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T2 := T2 + Hi (T1);
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if Hi (T2) /= 0 then
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Q := 0;
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R := X;
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return;
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end if;
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Du := Lo (T2) & Lo (T1);
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-- Set final signs (RM 4.5.5(27-30))
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Den_Pos := (Y < 0) = (Z < 0);
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-- Check overflow case of largest negative number divided by 1
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if X = Int64'First and then Du = 1 and then not Den_Pos then
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Raise_Error;
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end if;
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-- Perform the actual division
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Qu := Xu / Du;
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Ru := Xu rem Du;
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-- Deal with rounding case
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if Round and then Ru > (Du - Uns64'(1)) / Uns64'(2) then
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Qu := Qu + Uns64'(1);
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end if;
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-- Case of dividend (X) sign positive
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if X >= 0 then
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R := To_Int (Ru);
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Q := (if Den_Pos then To_Int (Qu) else -To_Int (Qu));
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-- Case of dividend (X) sign negative
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else
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R := -To_Int (Ru);
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Q := (if Den_Pos then -To_Int (Qu) else To_Int (Qu));
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end if;
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end Double_Divide;
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---------
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-- Le3 --
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---------
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function Le3 (X1, X2, X3 : Uns32; Y1, Y2, Y3 : Uns32) return Boolean is
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begin
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if X1 < Y1 then
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return True;
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elsif X1 > Y1 then
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return False;
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elsif X2 < Y2 then
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return True;
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elsif X2 > Y2 then
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return False;
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else
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return X3 <= Y3;
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end if;
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end Le3;
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-------------------------------
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-- Multiply_With_Ovflo_Check --
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-------------------------------
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function Multiply_With_Ovflo_Check (X, Y : Int64) return Int64 is
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Xu : constant Uns64 := abs X;
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Xhi : constant Uns32 := Hi (Xu);
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Xlo : constant Uns32 := Lo (Xu);
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Yu : constant Uns64 := abs Y;
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Yhi : constant Uns32 := Hi (Yu);
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Ylo : constant Uns32 := Lo (Yu);
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T1, T2 : Uns64;
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begin
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if Xhi /= 0 then
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if Yhi /= 0 then
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Raise_Error;
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else
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T2 := Xhi * Ylo;
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end if;
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elsif Yhi /= 0 then
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T2 := Xlo * Yhi;
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else -- Yhi = Xhi = 0
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T2 := 0;
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end if;
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-- Here we have T2 set to the contribution to the upper half of the
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-- result from the upper halves of the input values.
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T1 := Xlo * Ylo;
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T2 := T2 + Hi (T1);
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if Hi (T2) /= 0 then
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Raise_Error;
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end if;
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T2 := Lo (T2) & Lo (T1);
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if X >= 0 then
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if Y >= 0 then
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return To_Pos_Int (T2);
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else
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return To_Neg_Int (T2);
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end if;
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else -- X < 0
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if Y < 0 then
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return To_Pos_Int (T2);
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else
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return To_Neg_Int (T2);
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end if;
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end if;
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end Multiply_With_Ovflo_Check;
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-----------------
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-- Raise_Error --
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-----------------
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procedure Raise_Error is
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begin
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raise Constraint_Error with "64-bit arithmetic overflow";
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end Raise_Error;
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-------------------
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-- Scaled_Divide --
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-------------------
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procedure Scaled_Divide
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(X, Y, Z : Int64;
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Q, R : out Int64;
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Round : Boolean)
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is
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Xu : constant Uns64 := abs X;
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Xhi : constant Uns32 := Hi (Xu);
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Xlo : constant Uns32 := Lo (Xu);
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Yu : constant Uns64 := abs Y;
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Yhi : constant Uns32 := Hi (Yu);
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Ylo : constant Uns32 := Lo (Yu);
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Zu : Uns64 := abs Z;
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Zhi : Uns32 := Hi (Zu);
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Zlo : Uns32 := Lo (Zu);
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D : array (1 .. 4) of Uns32;
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-- The dividend, four digits (D(1) is high order)
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Qd : array (1 .. 2) of Uns32;
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-- The quotient digits, two digits (Qd(1) is high order)
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S1, S2, S3 : Uns32;
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-- Value to subtract, three digits (S1 is high order)
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Qu : Uns64;
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Ru : Uns64;
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-- Unsigned quotient and remainder
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Scale : Natural;
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-- Scaling factor used for multiple-precision divide. Dividend and
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-- Divisor are multiplied by 2 ** Scale, and the final remainder is
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-- divided by the scaling factor. The reason for this scaling is to
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-- allow more accurate estimation of quotient digits.
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T1, T2, T3 : Uns64;
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-- Temporary values
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begin
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-- First do the multiplication, giving the four digit dividend
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T1 := Xlo * Ylo;
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D (4) := Lo (T1);
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D (3) := Hi (T1);
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if Yhi /= 0 then
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T1 := Xlo * Yhi;
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T2 := D (3) + Lo (T1);
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D (3) := Lo (T2);
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D (2) := Hi (T1) + Hi (T2);
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if Xhi /= 0 then
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T1 := Xhi * Ylo;
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T2 := D (3) + Lo (T1);
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D (3) := Lo (T2);
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T3 := D (2) + Hi (T1);
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T3 := T3 + Hi (T2);
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D (2) := Lo (T3);
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D (1) := Hi (T3);
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T1 := (D (1) & D (2)) + Uns64'(Xhi * Yhi);
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D (1) := Hi (T1);
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D (2) := Lo (T1);
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else
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D (1) := 0;
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end if;
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else
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if Xhi /= 0 then
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T1 := Xhi * Ylo;
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T2 := D (3) + Lo (T1);
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D (3) := Lo (T2);
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D (2) := Hi (T1) + Hi (T2);
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else
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D (2) := 0;
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end if;
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D (1) := 0;
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end if;
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-- Now it is time for the dreaded multiple precision division. First an
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-- easy case, check for the simple case of a one digit divisor.
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if Zhi = 0 then
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if D (1) /= 0 or else D (2) >= Zlo then
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Raise_Error;
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-- Here we are dividing at most three digits by one digit
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else
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T1 := D (2) & D (3);
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T2 := Lo (T1 rem Zlo) & D (4);
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Qu := Lo (T1 / Zlo) & Lo (T2 / Zlo);
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Ru := T2 rem Zlo;
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end if;
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-- If divisor is double digit and too large, raise error
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elsif (D (1) & D (2)) >= Zu then
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Raise_Error;
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-- This is the complex case where we definitely have a double digit
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-- divisor and a dividend of at least three digits. We use the classical
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-- multiple division algorithm (see section (4.3.1) of Knuth's "The Art
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-- of Computer Programming", Vol. 2 for a description (algorithm D).
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else
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-- First normalize the divisor so that it has the leading bit on.
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-- We do this by finding the appropriate left shift amount.
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Scale := 0;
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if (Zhi and 16#FFFF0000#) = 0 then
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Scale := 16;
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Zu := Shift_Left (Zu, 16);
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end if;
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if (Hi (Zu) and 16#FF00_0000#) = 0 then
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Scale := Scale + 8;
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Zu := Shift_Left (Zu, 8);
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end if;
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if (Hi (Zu) and 16#F000_0000#) = 0 then
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Scale := Scale + 4;
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Zu := Shift_Left (Zu, 4);
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end if;
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if (Hi (Zu) and 16#C000_0000#) = 0 then
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Scale := Scale + 2;
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Zu := Shift_Left (Zu, 2);
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end if;
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if (Hi (Zu) and 16#8000_0000#) = 0 then
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Scale := Scale + 1;
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Zu := Shift_Left (Zu, 1);
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end if;
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Zhi := Hi (Zu);
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Zlo := Lo (Zu);
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-- Note that when we scale up the dividend, it still fits in four
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-- digits, since we already tested for overflow, and scaling does
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-- not change the invariant that (D (1) & D (2)) >= Zu.
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T1 := Shift_Left (D (1) & D (2), Scale);
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D (1) := Hi (T1);
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T2 := Shift_Left (0 & D (3), Scale);
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D (2) := Lo (T1) or Hi (T2);
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T3 := Shift_Left (0 & D (4), Scale);
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D (3) := Lo (T2) or Hi (T3);
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D (4) := Lo (T3);
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-- Loop to compute quotient digits, runs twice for Qd(1) and Qd(2)
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for J in 0 .. 1 loop
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-- Compute next quotient digit. We have to divide three digits by
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-- two digits. We estimate the quotient by dividing the leading
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-- two digits by the leading digit. Given the scaling we did above
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-- which ensured the first bit of the divisor is set, this gives
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-- an estimate of the quotient that is at most two too high.
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Qd (J + 1) := (if D (J + 1) = Zhi
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then 2 ** 32 - 1
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else Lo ((D (J + 1) & D (J + 2)) / Zhi));
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-- Compute amount to subtract
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T1 := Qd (J + 1) * Zlo;
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T2 := Qd (J + 1) * Zhi;
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S3 := Lo (T1);
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T1 := Hi (T1) + Lo (T2);
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S2 := Lo (T1);
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S1 := Hi (T1) + Hi (T2);
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-- Adjust quotient digit if it was too high
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loop
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exit when Le3 (S1, S2, S3, D (J + 1), D (J + 2), D (J + 3));
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Qd (J + 1) := Qd (J + 1) - 1;
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Sub3 (S1, S2, S3, 0, Zhi, Zlo);
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end loop;
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-- Now subtract S1&S2&S3 from D1&D2&D3 ready for next step
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Sub3 (D (J + 1), D (J + 2), D (J + 3), S1, S2, S3);
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end loop;
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-- The two quotient digits are now set, and the remainder of the
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-- scaled division is in D3&D4. To get the remainder for the
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-- original unscaled division, we rescale this dividend.
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-- We rescale the divisor as well, to make the proper comparison
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-- for rounding below.
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Qu := Qd (1) & Qd (2);
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Ru := Shift_Right (D (3) & D (4), Scale);
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Zu := Shift_Right (Zu, Scale);
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end if;
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-- Deal with rounding case
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if Round and then Ru > (Zu - Uns64'(1)) / Uns64'(2) then
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Qu := Qu + Uns64 (1);
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end if;
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-- Set final signs (RM 4.5.5(27-30))
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-- Case of dividend (X * Y) sign positive
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if (X >= 0 and then Y >= 0) or else (X < 0 and then Y < 0) then
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R := To_Pos_Int (Ru);
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Q := (if Z > 0 then To_Pos_Int (Qu) else To_Neg_Int (Qu));
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-- Case of dividend (X * Y) sign negative
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else
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R := To_Neg_Int (Ru);
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Q := (if Z > 0 then To_Neg_Int (Qu) else To_Pos_Int (Qu));
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end if;
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end Scaled_Divide;
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----------
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-- Sub3 --
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----------
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procedure Sub3 (X1, X2, X3 : in out Uns32; Y1, Y2, Y3 : Uns32) is
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begin
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if Y3 > X3 then
|
|
if X2 = 0 then
|
|
X1 := X1 - 1;
|
|
end if;
|
|
|
|
X2 := X2 - 1;
|
|
end if;
|
|
|
|
X3 := X3 - Y3;
|
|
|
|
if Y2 > X2 then
|
|
X1 := X1 - 1;
|
|
end if;
|
|
|
|
X2 := X2 - Y2;
|
|
X1 := X1 - Y1;
|
|
end Sub3;
|
|
|
|
-------------------------------
|
|
-- Subtract_With_Ovflo_Check --
|
|
-------------------------------
|
|
|
|
function Subtract_With_Ovflo_Check (X, Y : Int64) return Int64 is
|
|
R : constant Int64 := To_Int (To_Uns (X) - To_Uns (Y));
|
|
|
|
begin
|
|
if X >= 0 then
|
|
if Y > 0 or else R >= 0 then
|
|
return R;
|
|
end if;
|
|
|
|
else -- X < 0
|
|
if Y <= 0 or else R < 0 then
|
|
return R;
|
|
end if;
|
|
end if;
|
|
|
|
Raise_Error;
|
|
end Subtract_With_Ovflo_Check;
|
|
|
|
----------------
|
|
-- To_Neg_Int --
|
|
----------------
|
|
|
|
function To_Neg_Int (A : Uns64) return Int64 is
|
|
R : constant Int64 := (if A = 2**63 then Int64'First else -To_Int (A));
|
|
-- Note that we can't just use the expression of the Else, because it
|
|
-- overflows for A = 2**63.
|
|
begin
|
|
if R <= 0 then
|
|
return R;
|
|
else
|
|
Raise_Error;
|
|
end if;
|
|
end To_Neg_Int;
|
|
|
|
----------------
|
|
-- To_Pos_Int --
|
|
----------------
|
|
|
|
function To_Pos_Int (A : Uns64) return Int64 is
|
|
R : constant Int64 := To_Int (A);
|
|
begin
|
|
if R >= 0 then
|
|
return R;
|
|
else
|
|
Raise_Error;
|
|
end if;
|
|
end To_Pos_Int;
|
|
|
|
end System.Arith_64;
|