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------------------------------------------------------------------------------ -- -- -- GNAT RUN-TIME COMPONENTS -- -- -- -- S Y S T E M . A R I T H _ 6 4 -- -- -- -- B o d y -- -- -- -- Copyright (C) 1992-2009, 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. -- -- -- ------------------------------------------------------------------------------ with Interfaces; use Interfaces; with Ada.Unchecked_Conversion; package body System.Arith_64 is pragma Suppress (Overflow_Check); pragma Suppress (Range_Check); subtype Uns64 is Unsigned_64; function To_Uns is new Ada.Unchecked_Conversion (Int64, Uns64); function To_Int is new Ada.Unchecked_Conversion (Uns64, Int64); subtype Uns32 is Unsigned_32; ----------------------- -- Local Subprograms -- ----------------------- function "+" (A, B : Uns32) return Uns64; function "+" (A : Uns64; B : Uns32) return Uns64; pragma Inline ("+"); -- Length doubling additions function "*" (A, B : Uns32) return Uns64; pragma Inline ("*"); -- Length doubling multiplication function "/" (A : Uns64; B : Uns32) return Uns64; pragma Inline ("/"); -- Length doubling division function "rem" (A : Uns64; B : Uns32) return Uns64; pragma Inline ("rem"); -- Length doubling remainder function "&" (Hi, Lo : Uns32) return Uns64; pragma Inline ("&"); -- Concatenate hi, lo values to form 64-bit result function Le3 (X1, X2, X3 : Uns32; Y1, Y2, Y3 : Uns32) return Boolean; -- Determines if 96 bit value X1&X2&X3 <= Y1&Y2&Y3 function Lo (A : Uns64) return Uns32; pragma Inline (Lo); -- Low order half of 64-bit value function Hi (A : Uns64) return Uns32; pragma Inline (Hi); -- High order half of 64-bit value procedure Sub3 (X1, X2, X3 : in out Uns32; Y1, Y2, Y3 : Uns32); -- Computes X1&X2&X3 := X1&X2&X3 - Y1&Y1&Y3 with mod 2**96 wrap function To_Neg_Int (A : Uns64) return Int64; -- Convert to negative integer equivalent. If the input is in the range -- 0 .. 2 ** 63, then the corresponding negative signed integer (obtained -- by negating the given value) is returned, otherwise constraint error -- is raised. function To_Pos_Int (A : Uns64) return Int64; -- Convert to positive integer equivalent. If the input is in the range -- 0 .. 2 ** 63-1, then the corresponding non-negative signed integer is -- returned, otherwise constraint error is raised. procedure Raise_Error; pragma No_Return (Raise_Error); -- Raise constraint error with appropriate message --------- -- "&" -- --------- function "&" (Hi, Lo : Uns32) return Uns64 is begin return Shift_Left (Uns64 (Hi), 32) or Uns64 (Lo); end "&"; --------- -- "*" -- --------- function "*" (A, B : Uns32) return Uns64 is begin return Uns64 (A) * Uns64 (B); end "*"; --------- -- "+" -- --------- function "+" (A, B : Uns32) return Uns64 is begin return Uns64 (A) + Uns64 (B); end "+"; function "+" (A : Uns64; B : Uns32) return Uns64 is begin return A + Uns64 (B); end "+"; --------- -- "/" -- --------- function "/" (A : Uns64; B : Uns32) return Uns64 is begin return A / Uns64 (B); end "/"; ----------- -- "rem" -- ----------- function "rem" (A : Uns64; B : Uns32) return Uns64 is begin return A rem Uns64 (B); end "rem"; -------------------------- -- Add_With_Ovflo_Check -- -------------------------- function Add_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 Add_With_Ovflo_Check; ------------------- -- Double_Divide -- ------------------- procedure Double_Divide (X, Y, Z : Int64; Q, R : out Int64; Round : Boolean) is Xu : constant Uns64 := To_Uns (abs X); Yu : constant Uns64 := To_Uns (abs Y); Yhi : constant Uns32 := Hi (Yu); Ylo : constant Uns32 := Lo (Yu); Zu : constant Uns64 := To_Uns (abs Z); Zhi : constant Uns32 := Hi (Zu); Zlo : constant Uns32 := Lo (Zu); T1, T2 : Uns64; Du, Qu, Ru : Uns64; Den_Pos : Boolean; begin if Yu = 0 or else Zu = 0 then Raise_Error; end if; -- Compute Y * Z. Note that if the result overflows 64 bits unsigned, -- then the rounded result is clearly zero (since the dividend is at -- most 2**63 - 1, the extra bit of precision is nice here!) if Yhi /= 0 then if Zhi /= 0 then Q := 0; R := X; return; else T2 := Yhi * Zlo; end if; else T2 := (if Zhi /= 0 then Ylo * Zhi else 0); end if; T1 := Ylo * Zlo; T2 := T2 + Hi (T1); if Hi (T2) /= 0 then Q := 0; R := X; return; end if; Du := Lo (T2) & Lo (T1); -- Set final signs (RM 4.5.5(27-30)) Den_Pos := (Y < 0) = (Z < 0); -- Check overflow case of largest negative number divided by 1 if X = Int64'First and then Du = 1 and then not Den_Pos then Raise_Error; end if; -- Perform the actual division Qu := Xu / Du; Ru := Xu rem Du; -- Deal with rounding case if Round and then Ru > (Du - Uns64'(1)) / Uns64'(2) then Qu := Qu + Uns64'(1); end if; -- Case of dividend (X) sign positive if X >= 0 then R := To_Int (Ru); Q := (if Den_Pos then To_Int (Qu) else -To_Int (Qu)); -- Case of dividend (X) sign negative else R := -To_Int (Ru); Q := (if Den_Pos then -To_Int (Qu) else To_Int (Qu)); end if; end Double_Divide; -------- -- Hi -- -------- function Hi (A : Uns64) return Uns32 is begin return Uns32 (Shift_Right (A, 32)); end Hi; --------- -- Le3 -- --------- function Le3 (X1, X2, X3 : Uns32; Y1, Y2, Y3 : Uns32) return Boolean is begin if X1 < Y1 then return True; elsif X1 > Y1 then return False; elsif X2 < Y2 then return True; elsif X2 > Y2 then return False; else return X3 <= Y3; end if; end Le3; -------- -- Lo -- -------- function Lo (A : Uns64) return Uns32 is begin return Uns32 (A and 16#FFFF_FFFF#); end Lo; ------------------------------- -- Multiply_With_Ovflo_Check -- ------------------------------- function Multiply_With_Ovflo_Check (X, Y : Int64) return Int64 is Xu : constant Uns64 := To_Uns (abs X); Xhi : constant Uns32 := Hi (Xu); Xlo : constant Uns32 := Lo (Xu); Yu : constant Uns64 := To_Uns (abs Y); Yhi : constant Uns32 := Hi (Yu); Ylo : constant Uns32 := Lo (Yu); T1, T2 : Uns64; begin if Xhi /= 0 then if Yhi /= 0 then Raise_Error; else T2 := Xhi * Ylo; end if; elsif Yhi /= 0 then T2 := Xlo * Yhi; else -- Yhi = Xhi = 0 T2 := 0; end if; -- Here we have T2 set to the contribution to the upper half -- of the result from the upper halves of the input values. T1 := Xlo * Ylo; T2 := T2 + Hi (T1); if Hi (T2) /= 0 then Raise_Error; end if; T2 := Lo (T2) & Lo (T1); if X >= 0 then if Y >= 0 then return To_Pos_Int (T2); else return To_Neg_Int (T2); end if; else -- X < 0 if Y < 0 then return To_Pos_Int (T2); else return To_Neg_Int (T2); end if; end if; end Multiply_With_Ovflo_Check; ----------------- -- Raise_Error -- ----------------- procedure Raise_Error is begin raise Constraint_Error with "64-bit arithmetic overflow"; end Raise_Error; ------------------- -- Scaled_Divide -- ------------------- procedure Scaled_Divide (X, Y, Z : Int64; Q, R : out Int64; Round : Boolean) is Xu : constant Uns64 := To_Uns (abs X); Xhi : constant Uns32 := Hi (Xu); Xlo : constant Uns32 := Lo (Xu); Yu : constant Uns64 := To_Uns (abs Y); Yhi : constant Uns32 := Hi (Yu); Ylo : constant Uns32 := Lo (Yu); Zu : Uns64 := To_Uns (abs Z); Zhi : Uns32 := Hi (Zu); Zlo : Uns32 := Lo (Zu); D : array (1 .. 4) of Uns32; -- The dividend, four digits (D(1) is high order) Qd : array (1 .. 2) of Uns32; -- The quotient digits, two digits (Qd(1) is high order) S1, S2, S3 : Uns32; -- Value to subtract, three digits (S1 is high order) Qu : Uns64; Ru : Uns64; -- Unsigned quotient and remainder Scale : Natural; -- Scaling factor used for multiple-precision divide. Dividend and -- Divisor are multiplied by 2 ** Scale, and the final remainder -- is divided by the scaling factor. The reason for this scaling -- is to allow more accurate estimation of quotient digits. T1, T2, T3 : Uns64; -- Temporary values begin -- First do the multiplication, giving the four digit dividend T1 := Xlo * Ylo; D (4) := Lo (T1); D (3) := Hi (T1); if Yhi /= 0 then T1 := Xlo * Yhi; T2 := D (3) + Lo (T1); D (3) := Lo (T2); D (2) := Hi (T1) + Hi (T2); if Xhi /= 0 then T1 := Xhi * Ylo; T2 := D (3) + Lo (T1); D (3) := Lo (T2); T3 := D (2) + Hi (T1); T3 := T3 + Hi (T2); D (2) := Lo (T3); D (1) := Hi (T3); T1 := (D (1) & D (2)) + Uns64'(Xhi * Yhi); D (1) := Hi (T1); D (2) := Lo (T1); else D (1) := 0; end if; else if Xhi /= 0 then T1 := Xhi * Ylo; T2 := D (3) + Lo (T1); D (3) := Lo (T2); D (2) := Hi (T1) + Hi (T2); else D (2) := 0; end if; D (1) := 0; end if; -- Now it is time for the dreaded multiple precision division. First -- an easy case, check for the simple case of a one digit divisor. if Zhi = 0 then if D (1) /= 0 or else D (2) >= Zlo then Raise_Error; -- Here we are dividing at most three digits by one digit else T1 := D (2) & D (3); T2 := Lo (T1 rem Zlo) & D (4); Qu := Lo (T1 / Zlo) & Lo (T2 / Zlo); Ru := T2 rem Zlo; end if; -- If divisor is double digit and too large, raise error elsif (D (1) & D (2)) >= Zu then Raise_Error; -- This is the complex case where we definitely have a double digit -- divisor and a dividend of at least three digits. We use the classical -- multiple division algorithm (see section (4.3.1) of Knuth's "The Art -- of Computer Programming", Vol. 2 for a description (algorithm D). else -- First normalize the divisor so that it has the leading bit on. -- We do this by finding the appropriate left shift amount. Scale := 0; if (Zhi and 16#FFFF0000#) = 0 then Scale := 16; Zu := Shift_Left (Zu, 16); end if; if (Hi (Zu) and 16#FF00_0000#) = 0 then Scale := Scale + 8; Zu := Shift_Left (Zu, 8); end if; if (Hi (Zu) and 16#F000_0000#) = 0 then Scale := Scale + 4; Zu := Shift_Left (Zu, 4); end if; if (Hi (Zu) and 16#C000_0000#) = 0 then Scale := Scale + 2; Zu := Shift_Left (Zu, 2); end if; if (Hi (Zu) and 16#8000_0000#) = 0 then Scale := Scale + 1; Zu := Shift_Left (Zu, 1); end if; Zhi := Hi (Zu); Zlo := Lo (Zu); -- Note that when we scale up the dividend, it still fits in four -- digits, since we already tested for overflow, and scaling does -- not change the invariant that (D (1) & D (2)) >= Zu. T1 := Shift_Left (D (1) & D (2), Scale); D (1) := Hi (T1); T2 := Shift_Left (0 & D (3), Scale); D (2) := Lo (T1) or Hi (T2); T3 := Shift_Left (0 & D (4), Scale); D (3) := Lo (T2) or Hi (T3); D (4) := Lo (T3); -- Loop to compute quotient digits, runs twice for Qd(1) and Qd(2) for J in 0 .. 1 loop -- Compute next quotient digit. We have to divide three digits by -- two digits. We estimate the quotient by dividing the leading -- two digits by the leading digit. Given the scaling we did above -- which ensured the first bit of the divisor is set, this gives -- an estimate of the quotient that is at most two too high. Qd (J + 1) := (if D (J + 1) = Zhi then 2 ** 32 - 1 else Lo ((D (J + 1) & D (J + 2)) / Zhi)); -- Compute amount to subtract T1 := Qd (J + 1) * Zlo; T2 := Qd (J + 1) * Zhi; S3 := Lo (T1); T1 := Hi (T1) + Lo (T2); S2 := Lo (T1); S1 := Hi (T1) + Hi (T2); -- Adjust quotient digit if it was too high loop exit when Le3 (S1, S2, S3, D (J + 1), D (J + 2), D (J + 3)); Qd (J + 1) := Qd (J + 1) - 1; Sub3 (S1, S2, S3, 0, Zhi, Zlo); end loop; -- Now subtract S1&S2&S3 from D1&D2&D3 ready for next step Sub3 (D (J + 1), D (J + 2), D (J + 3), S1, S2, S3); end loop; -- The two quotient digits are now set, and the remainder of the -- scaled division is in D3&D4. To get the remainder for the -- original unscaled division, we rescale this dividend. -- We rescale the divisor as well, to make the proper comparison -- for rounding below. Qu := Qd (1) & Qd (2); Ru := Shift_Right (D (3) & D (4), Scale); Zu := Shift_Right (Zu, Scale); end if; -- Deal with rounding case if Round and then Ru > (Zu - Uns64'(1)) / Uns64'(2) then Qu := Qu + Uns64 (1); end if; -- Set final signs (RM 4.5.5(27-30)) -- Case of dividend (X * Y) sign positive if (X >= 0 and then Y >= 0) or else (X < 0 and then Y < 0) then R := To_Pos_Int (Ru); Q := (if Z > 0 then To_Pos_Int (Qu) else To_Neg_Int (Qu)); -- Case of dividend (X * Y) sign negative else R := To_Neg_Int (Ru); Q := (if Z > 0 then To_Neg_Int (Qu) else To_Pos_Int (Qu)); end if; end Scaled_Divide; ---------- -- Sub3 -- ---------- procedure Sub3 (X1, X2, X3 : in out Uns32; Y1, Y2, Y3 : Uns32) is begin 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 := -To_Int (A); 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;
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