------------------------------------------------------------------------------ -- -- -- GNAT RUN-TIME COMPONENTS -- -- -- -- S Y S T E M . A R I T H _ D O U B L E -- -- -- -- B o d y -- -- -- -- Copyright (C) 1992-2020, 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. -- -- -- ------------------------------------------------------------------------------ with Ada.Unchecked_Conversion; package body System.Arith_Double is pragma Suppress (Overflow_Check); pragma Suppress (Range_Check); function To_Uns is new Ada.Unchecked_Conversion (Double_Int, Double_Uns); function To_Int is new Ada.Unchecked_Conversion (Double_Uns, Double_Int); Double_Size : constant Natural := Double_Int'Size; Single_Size : constant Natural := Double_Int'Size / 2; ----------------------- -- Local Subprograms -- ----------------------- function "+" (A, B : Single_Uns) return Double_Uns is (Double_Uns (A) + Double_Uns (B)); function "+" (A : Double_Uns; B : Single_Uns) return Double_Uns is (A + Double_Uns (B)); -- Length doubling additions function "*" (A, B : Single_Uns) return Double_Uns is (Double_Uns (A) * Double_Uns (B)); -- Length doubling multiplication function "/" (A : Double_Uns; B : Single_Uns) return Double_Uns is (A / Double_Uns (B)); -- Length doubling division function "&" (Hi, Lo : Single_Uns) return Double_Uns is (Shift_Left (Double_Uns (Hi), Single_Size) or Double_Uns (Lo)); -- Concatenate hi, lo values to form double result function "abs" (X : Double_Int) return Double_Uns is (if X = Double_Int'First then 2 ** (Double_Size - 1) else Double_Uns (Double_Int'(abs X))); -- Convert absolute value of X to unsigned. Note that we can't just use -- the expression of the Else since it overflows for X = Double_Int'First. function "rem" (A : Double_Uns; B : Single_Uns) return Double_Uns is (A rem Double_Uns (B)); -- Length doubling remainder function Le3 (X1, X2, X3, Y1, Y2, Y3 : Single_Uns) return Boolean; -- Determines if (3 * Single_Size)-bit value X1&X2&X3 <= Y1&Y2&Y3 function Lo (A : Double_Uns) return Single_Uns is (Single_Uns (A and (2 ** Single_Size - 1))); -- Low order half of double value function Hi (A : Double_Uns) return Single_Uns is (Single_Uns (Shift_Right (A, Single_Size))); -- High order half of double value procedure Sub3 (X1, X2, X3 : in out Single_Uns; Y1, Y2, Y3 : Single_Uns); -- Computes X1&X2&X3 := X1&X2&X3 - Y1&Y1&Y3 mod 2 ** (3 * Single_Size) function To_Neg_Int (A : Double_Uns) return Double_Int; -- Convert to negative integer equivalent. If the input is in the range -- 0 .. 2 ** (Double_Size - 1), then the corresponding nonpositive signed -- integer (obtained by negating the given value) is returned, otherwise -- constraint error is raised. function To_Pos_Int (A : Double_Uns) return Double_Int; -- Convert to positive integer equivalent. If the input is in the range -- 0 .. 2 ** (Double_Size - 1) - 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 -------------------------- -- Add_With_Ovflo_Check -- -------------------------- function Add_With_Ovflo_Check (X, Y : Double_Int) return Double_Int is R : constant Double_Int := 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 : Double_Int; Q, R : out Double_Int; Round : Boolean) is Xu : constant Double_Uns := abs X; Yu : constant Double_Uns := abs Y; Yhi : constant Single_Uns := Hi (Yu); Ylo : constant Single_Uns := Lo (Yu); Zu : constant Double_Uns := abs Z; Zhi : constant Single_Uns := Hi (Zu); Zlo : constant Single_Uns := Lo (Zu); T1, T2 : Double_Uns; Du, Qu, Ru : Double_Uns; Den_Pos : Boolean; begin if Yu = 0 or else Zu = 0 then Raise_Error; end if; -- Set final signs (RM 4.5.5(27-30)) Den_Pos := (Y < 0) = (Z < 0); -- Compute Y * Z. Note that if the result overflows Double_Uns, then -- the rounded result is zero, except for the very special case where -- X = -2 ** (Double_Size - 1) and abs(Y*Z) = 2 ** Double_Size, when -- Round is True. if Yhi /= 0 then if Zhi /= 0 then -- Handle the special case when Round is True if Yhi = 1 and then Zhi = 1 and then Ylo = 0 and then Zlo = 0 and then X = Double_Int'First and then Round then Q := (if Den_Pos then -1 else 1); else Q := 0; end if; R := X; return; else T2 := Yhi * Zlo; end if; else T2 := Ylo * Zhi; end if; T1 := Ylo * Zlo; T2 := T2 + Hi (T1); if Hi (T2) /= 0 then -- Handle the special case when Round is True if Hi (T2) = 1 and then Lo (T2) = 0 and then Lo (T1) = 0 and then X = Double_Int'First and then Round then Q := (if Den_Pos then -1 else 1); else Q := 0; end if; R := X; return; end if; Du := Lo (T2) & Lo (T1); -- Check overflow case of largest negative number divided by -1 if X = Double_Int'First and then Du = 1 and then not Den_Pos then Raise_Error; end if; -- Perform the actual division pragma Assert (Du /= 0); -- Multiplication of 2-limb arguments Yu and Zu leads to 4-limb result -- (where each limb is a single value). Cases where 4 limbs are needed -- require Yhi/=0 and Zhi/=0 and lead to early exit. Remaining cases -- where 3 limbs are needed correspond to Hi(T2)/=0 and lead to early -- exit. Thus, at this point, the result fits in 2 limbs which are -- exactly Lo(T2) and Lo(T1), which corresponds to the value of Du. -- As the case where one of Yu or Zu is null also led to early exit, -- we have Du/=0 here. Qu := Xu / Du; Ru := Xu rem Du; -- Deal with rounding case if Round and then Ru > (Du - Double_Uns'(1)) / Double_Uns'(2) then Qu := Qu + Double_Uns'(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 -- We perform the unary minus operation on the unsigned value -- before conversion to signed, to avoid a possible overflow -- for value -2 ** (Double_Size - 1), both for computing R and Q. else R := To_Int (-Ru); Q := (if Den_Pos then To_Int (-Qu) else To_Int (Qu)); end if; end Double_Divide; --------- -- Le3 -- --------- function Le3 (X1, X2, X3, Y1, Y2, Y3 : Single_Uns) 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; ------------------------------- -- Multiply_With_Ovflo_Check -- ------------------------------- function Multiply_With_Ovflo_Check (X, Y : Double_Int) return Double_Int is Xu : constant Double_Uns := abs X; Xhi : constant Single_Uns := Hi (Xu); Xlo : constant Single_Uns := Lo (Xu); Yu : constant Double_Uns := abs Y; Yhi : constant Single_Uns := Hi (Yu); Ylo : constant Single_Uns := Lo (Yu); T1, T2 : Double_Uns; 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); pragma Annotate (CodePeer, Intentional, "precondition", "Intentional Unsigned->Signed conversion"); else return To_Neg_Int (T2); end if; else -- X < 0 if Y < 0 then return To_Pos_Int (T2); pragma Annotate (CodePeer, Intentional, "precondition", "Intentional Unsigned->Signed conversion"); 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 "Double arithmetic overflow"; end Raise_Error; ------------------- -- Scaled_Divide -- ------------------- procedure Scaled_Divide (X, Y, Z : Double_Int; Q, R : out Double_Int; Round : Boolean) is Xu : constant Double_Uns := abs X; Xhi : constant Single_Uns := Hi (Xu); Xlo : constant Single_Uns := Lo (Xu); Yu : constant Double_Uns := abs Y; Yhi : constant Single_Uns := Hi (Yu); Ylo : constant Single_Uns := Lo (Yu); Zu : Double_Uns := abs Z; Zhi : Single_Uns := Hi (Zu); Zlo : Single_Uns := Lo (Zu); D : array (1 .. 4) of Single_Uns; -- The dividend, four digits (D(1) is high order) Qd : array (1 .. 2) of Single_Uns; -- The quotient digits, two digits (Qd(1) is high order) S1, S2, S3 : Single_Uns; -- Value to subtract, three digits (S1 is high order) Qu : Double_Uns; Ru : Double_Uns; -- Unsigned quotient and remainder Mask : Single_Uns; -- Mask of bits used to compute the scaling factor below 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. Shift : Natural; -- Shift factor used to compute the scaling factor above T1, T2, T3 : Double_Uns; -- 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)) + Double_Uns'(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 dividend is 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-precision 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. Shift := Single_Size / 2; Mask := Shift_Left (2 ** (Single_Size / 2) - 1, Shift); Scale := 0; while Shift /= 0 loop if (Hi (Zu) and Mask) = 0 then Scale := Scale + Shift; Zu := Shift_Left (Zu, Shift); end if; Shift := Shift / 2; Mask := Shift_Left (Mask, Shift); end loop; Zhi := Hi (Zu); Zlo := Lo (Zu); pragma Assert (Zhi /= 0); -- We have Hi(Zu)/=0 before normalization. The sequence of Shift_Left -- operations results in the leading bit of Zu being 1 by moving the -- leftmost 1-bit in Zu to leading position, thus Zhi=Hi(Zu)/=0 here. -- 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 ** Single_Size - 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 -- We use the version of the algorithm in the 2nd Edition of -- "The Art of Computer Programming". This had a bug not -- discovered till 1995, see Vol 2 errata: -- http://www-cs-faculty.stanford.edu/~uno/err2-2e.ps.gz. -- Under rare circumstances the expression in the test could -- overflow. This version was further corrected in 2005, see -- Vol 2 errata: -- http://www-cs-faculty.stanford.edu/~uno/all2-pre.ps.gz. -- This implementation is not impacted by these bugs, due to the -- use of a word-size comparison done in function Le3 instead of -- a comparison on two-word integer quantities in the original -- algorithm. 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 - Double_Uns'(1)) / Double_Uns'(2) then -- Protect against wrapping around when rounding, by signaling -- an overflow when the quotient is too large. if Qu = Double_Uns'Last then Raise_Error; end if; Qu := Qu + Double_Uns'(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 Single_Uns; Y1, Y2, Y3 : Single_Uns) 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 : Double_Int) return Double_Int is R : constant Double_Int := 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 : Double_Uns) return Double_Int is R : constant Double_Int := (if A = 2 ** (Double_Size - 1) then Double_Int'First else -To_Int (A)); -- Note that we can't just use the expression of the Else, because it -- overflows for A = 2 ** (Double_Size - 1). begin if R <= 0 then return R; else Raise_Error; end if; end To_Neg_Int; ---------------- -- To_Pos_Int -- ---------------- function To_Pos_Int (A : Double_Uns) return Double_Int is R : constant Double_Int := To_Int (A); begin if R >= 0 then return R; else Raise_Error; end if; end To_Pos_Int; end System.Arith_Double;