------------------------------------------------------------------------------
-- --
-- GNAT RUN-TIME COMPONENTS --
-- --
-- S Y S T E M . F O R E _ F --
-- --
-- B o d y --
-- --
-- Copyright (C) 2020-2023, Free Software Foundation, Inc. --
-- --
-- GNAT is free software; you can redistribute it and/or modify it under --
-- terms of the GNU General Public License as published by the Free Soft- --
-- ware Foundation; either version 3, or (at your option) any later ver- --
-- sion. GNAT is distributed in the hope that it will be useful, but WITH- --
-- OUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY --
-- or FITNESS FOR A PARTICULAR PURPOSE. --
-- --
-- As a special exception under Section 7 of GPL version 3, you are granted --
-- additional permissions described in the GCC Runtime Library Exception, --
-- version 3.1, as published by the Free Software Foundation. --
-- --
-- You should have received a copy of the GNU General Public License and --
-- a copy of the GCC Runtime Library Exception along with this program; --
-- see the files COPYING3 and COPYING.RUNTIME respectively. If not, see --
-- <http://www.gnu.org/licenses/>. --
-- --
-- GNAT was originally developed by the GNAT team at New York University. --
-- Extensive contributions were provided by Ada Core Technologies Inc. --
-- --
------------------------------------------------------------------------------
package body System.Fore_F is
Maxdigs : constant Natural := Int'Width - 2;
-- Maximum number of decimal digits that can be represented in an Int.
-- The "-2" accounts for the sign and one extra digit, since we need the
-- maximum number of 9's that can be represented, e.g. for the 64-bit case,
-- Integer_64'Width is 20 since the maximum value is approximately 9.2E+18
-- and has 19 digits, but the maximum number of 9's that can be represented
-- in Integer_64 is only 18.
-- The first prerequisite of the implementation is that the scaled divide
-- does not overflow, which means that the absolute value of the bounds of
-- the subtype must be smaller than 10**Maxdigs * 2**(Int'Size - 1).
-- Otherwise Constraint_Error is raised by the scaled divide operation.
-- The second prerequisite is that the computation of the operands does not
-- overflow, which means that, if the small is larger than 1, it is either
-- an integer or its numerator and denominator must be both smaller than
-- the power 10**(Maxdigs - 1).
----------------
-- Fore_Fixed --
----------------
function Fore_Fixed (Lo, Hi, Num, Den : Int; Scale : Integer) return Natural
is
pragma Assert (Num < 0 and then Den < 0);
-- Accept only negative numbers to allow -2**(Int'Size - 1)
function Negative_Abs (Val : Int) return Int is
(if Val <= 0 then Val else -Val);
-- Return the opposite of the absolute value of Val
T : Int := Int'Min (Negative_Abs (Lo), Negative_Abs (Hi));
F : Natural;
Q, R : Int;
begin
-- Initial value of 2 allows for sign and mandatory single digit
F := 2;
-- The easy case is when Num is not larger than Den in magnitude,
-- i.e. if S = Num / Den, then S <= 1, in which case we can just
-- compute the product Q = T * S.
if Num >= Den then
Scaled_Divide (T, Num, Den, Q, R, Round => False);
T := Q;
-- Otherwise S > 1 and thus Scale <= 0, compute Q and R such that
-- T * Num = Q * (Den * 10**(-D)) + R
-- with
-- D = Integer'Max (-Maxdigs, Scale - 1)
-- then reason on Q if it is non-zero or else on R / Den.
-- This works only if Den * 10**(-D) does not overflow, which is true
-- if Den = 1. Suppose that Num corresponds to the maximum value of -D,
-- i.e. Maxdigs and 10**(-D) = 10**Maxdigs. If you change Den into 10,
-- then S becomes 10 times smaller and, therefore, Scale is incremented
-- by 1, which means that -D is decremented by 1 provided that Scale was
-- initially not smaller than 1 - Maxdigs, so the multiplication still
-- does not overflow. But you need to reach 10 to trigger this effect,
-- which means that a leeway of 10 is required, so let's restrict this
-- to a Num for which 10**(-D) <= 10**(Maxdigs - 1). To sum up, if S is
-- the ratio of two integers with
-- 1 < Den < Num <= B
-- where B is a fixed limit, then the multiplication does not overflow.
-- B can be taken as the largest integer Small such that D = 1 - Maxdigs
-- i.e. such that Scale = 2 - Maxdigs, which is 10**(Maxdigs - 1) - 1.
else
declare
D : constant Integer := Integer'Max (-Maxdigs, Scale - 1);
begin
Scaled_Divide (T, Num, Den * 10**(-D), Q, R, Round => False);
if Q /= 0 then
T := Q;
F := F - D;
else
T := R / Den;
end if;
end;
end if;
-- Loop to increase Fore as needed to include full range of values
while T <= -10 or else T >= 10 loop
T := T / 10;
F := F + 1;
end loop;
return F;
end Fore_Fixed;
end System.Fore_F;