------------------------------------------------------------------------------ -- -- -- GNAT RUN-TIME COMPONENTS -- -- -- -- S Y S T E M . F O R E _ F -- -- -- -- B o d y -- -- -- -- Copyright (C) 2020-2022, 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 -- -- . -- -- -- -- 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;