------------------------------------------------------------------------------ -- -- -- GNAT RUNTIME COMPONENTS -- -- -- -- A D A . N U M E R I C S . A U X -- -- -- -- B o d y -- -- (Machine Version for x86) -- -- -- -- -- -- Copyright (C) 1998-2001 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 2, 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. See the GNU General Public License -- -- for more details. You should have received a copy of the GNU General -- -- Public License distributed with GNAT; see file COPYING. If not, write -- -- to the Free Software Foundation, 59 Temple Place - Suite 330, Boston, -- -- MA 02111-1307, USA. -- -- -- -- As a special exception, if other files instantiate generics from this -- -- unit, or you link this unit with other files to produce an executable, -- -- this unit does not by itself cause the resulting executable to be -- -- covered by the GNU General Public License. This exception does not -- -- however invalidate any other reasons why the executable file might be -- -- covered by the GNU Public License. -- -- -- -- GNAT was originally developed by the GNAT team at New York University. -- -- Extensive contributions were provided by Ada Core Technologies Inc. -- -- -- ------------------------------------------------------------------------------ -- File a-numaux.adb <- 86numaux.adb -- This version of Numerics.Aux is for the IEEE Double Extended floating -- point format on x86. with System.Machine_Code; use System.Machine_Code; package body Ada.Numerics.Aux is NL : constant String := ASCII.LF & ASCII.HT; type FPU_Stack_Pointer is range 0 .. 7; for FPU_Stack_Pointer'Size use 3; type FPU_Status_Word is record B : Boolean; -- FPU Busy (for 8087 compatibility only) ES : Boolean; -- Error Summary Status SF : Boolean; -- Stack Fault Top : FPU_Stack_Pointer; -- Condition Code Flags -- C2 is set by FPREM and FPREM1 to indicate incomplete reduction. -- In case of successfull recorction, C0, C3 and C1 are set to the -- three least significant bits of the result (resp. Q2, Q1 and Q0). -- C2 is used by FPTAN, FSIN, FCOS, and FSINCOS to indicate that -- that source operand is beyond the allowable range of -- -2.0**63 .. 2.0**63. C3 : Boolean; C2 : Boolean; C1 : Boolean; C0 : Boolean; -- Exception Flags PE : Boolean; -- Precision UE : Boolean; -- Underflow OE : Boolean; -- Overflow ZE : Boolean; -- Zero Divide DE : Boolean; -- Denormalized Operand IE : Boolean; -- Invalid Operation end record; for FPU_Status_Word use record B at 0 range 15 .. 15; C3 at 0 range 14 .. 14; Top at 0 range 11 .. 13; C2 at 0 range 10 .. 10; C1 at 0 range 9 .. 9; C0 at 0 range 8 .. 8; ES at 0 range 7 .. 7; SF at 0 range 6 .. 6; PE at 0 range 5 .. 5; UE at 0 range 4 .. 4; OE at 0 range 3 .. 3; ZE at 0 range 2 .. 2; DE at 0 range 1 .. 1; IE at 0 range 0 .. 0; end record; for FPU_Status_Word'Size use 16; ----------------------- -- Local subprograms -- ----------------------- function Is_Nan (X : Double) return Boolean; -- Return True iff X is a IEEE NaN value function Logarithmic_Pow (X, Y : Double) return Double; -- Implementation of X**Y using Exp and Log functions (binary base) -- to calculate the exponentiation. This is used by Pow for values -- for values of Y in the open interval (-0.25, 0.25) function Reduce (X : Double) return Double; -- Implement partial reduction of X by Pi in the x86. -- Note that for the Sin, Cos and Tan functions completely accurate -- reduction of the argument is done for arguments in the range of -- -2.0**63 .. 2.0**63, using a 66-bit approximation of Pi. pragma Inline (Is_Nan); pragma Inline (Reduce); --------------------------------- -- Basic Elementary Functions -- --------------------------------- -- This section implements a few elementary functions that are -- used to build the more complex ones. This ordering enables -- better inlining. ---------- -- Atan -- ---------- function Atan (X : Double) return Double is Result : Double; begin Asm (Template => "fld1" & NL & "fpatan", Outputs => Double'Asm_Output ("=t", Result), Inputs => Double'Asm_Input ("0", X)); -- The result value is NaN iff input was invalid if not (Result = Result) then raise Argument_Error; end if; return Result; end Atan; --------- -- Exp -- --------- function Exp (X : Double) return Double is Result : Double; begin Asm (Template => "fldl2e " & NL & "fmulp %%st, %%st(1)" & NL -- X * log2 (E) & "fld %%st(0) " & NL & "frndint " & NL -- Integer (X * Log2 (E)) & "fsubr %%st, %%st(1)" & NL -- Fraction (X * Log2 (E)) & "fxch " & NL & "f2xm1 " & NL -- 2**(...) - 1 & "fld1 " & NL & "faddp %%st, %%st(1)" & NL -- 2**(Fraction (X * Log2 (E))) & "fscale " & NL -- E ** X & "fstp %%st(1) ", Outputs => Double'Asm_Output ("=t", Result), Inputs => Double'Asm_Input ("0", X)); return Result; end Exp; ------------ -- Is_Nan -- ------------ function Is_Nan (X : Double) return Boolean is begin -- The IEEE NaN values are the only ones that do not equal themselves return not (X = X); end Is_Nan; --------- -- Log -- --------- function Log (X : Double) return Double is Result : Double; begin Asm (Template => "fldln2 " & NL & "fxch " & NL & "fyl2x " & NL, Outputs => Double'Asm_Output ("=t", Result), Inputs => Double'Asm_Input ("0", X)); return Result; end Log; ------------ -- Reduce -- ------------ function Reduce (X : Double) return Double is Result : Double; begin Asm (Template => -- Partial argument reduction "fldpi " & NL & "fadd %%st(0), %%st" & NL & "fxch %%st(1) " & NL & "fprem1 " & NL & "fstp %%st(1) ", Outputs => Double'Asm_Output ("=t", Result), Inputs => Double'Asm_Input ("0", X)); return Result; end Reduce; ---------- -- Sqrt -- ---------- function Sqrt (X : Double) return Double is Result : Double; begin if X < 0.0 then raise Argument_Error; end if; Asm (Template => "fsqrt", Outputs => Double'Asm_Output ("=t", Result), Inputs => Double'Asm_Input ("0", X)); return Result; end Sqrt; --------------------------------- -- Other Elementary Functions -- --------------------------------- -- These are built using the previously implemented basic functions ---------- -- Acos -- ---------- function Acos (X : Double) return Double is Result : Double; begin Result := 2.0 * Atan (Sqrt ((1.0 - X) / (1.0 + X))); -- The result value is NaN iff input was invalid if Is_Nan (Result) then raise Argument_Error; end if; return Result; end Acos; ---------- -- Asin -- ---------- function Asin (X : Double) return Double is Result : Double; begin Result := Atan (X / Sqrt ((1.0 - X) * (1.0 + X))); -- The result value is NaN iff input was invalid if Is_Nan (Result) then raise Argument_Error; end if; return Result; end Asin; --------- -- Cos -- --------- function Cos (X : Double) return Double is Reduced_X : Double := X; Result : Double; Status : FPU_Status_Word; begin loop Asm (Template => "fcos " & NL & "xorl %%eax, %%eax " & NL & "fnstsw %%ax ", Outputs => (Double'Asm_Output ("=t", Result), FPU_Status_Word'Asm_Output ("=a", Status)), Inputs => Double'Asm_Input ("0", Reduced_X)); exit when not Status.C2; -- Original argument was not in range and the result -- is the unmodified argument. Reduced_X := Reduce (Result); end loop; return Result; end Cos; --------------------- -- Logarithmic_Pow -- --------------------- function Logarithmic_Pow (X, Y : Double) return Double is Result : Double; begin Asm (Template => "" -- X : Y & "fyl2x " & NL -- Y * Log2 (X) & "fst %%st(1) " & NL -- Y * Log2 (X) : Y * Log2 (X) & "frndint " & NL -- Int (...) : Y * Log2 (X) & "fsubr %%st, %%st(1)" & NL -- Int (...) : Fract (...) & "fxch " & NL -- Fract (...) : Int (...) & "f2xm1 " & NL -- 2**Fract (...) - 1 : Int (...) & "fld1 " & NL -- 1 : 2**Fract (...) - 1 : Int (...) & "faddp %%st, %%st(1)" & NL -- 2**Fract (...) : Int (...) & "fscale " & NL -- 2**(Fract (...) + Int (...)) & "fstp %%st(1) ", Outputs => Double'Asm_Output ("=t", Result), Inputs => (Double'Asm_Input ("0", X), Double'Asm_Input ("u", Y))); return Result; end Logarithmic_Pow; --------- -- Pow -- --------- function Pow (X, Y : Double) return Double is type Mantissa_Type is mod 2**Double'Machine_Mantissa; -- Modular type that can hold all bits of the mantissa of Double -- For negative exponents, a division is done -- at the end of the processing. Negative_Y : constant Boolean := Y < 0.0; Abs_Y : constant Double := abs Y; -- During this function the following invariant is kept: -- X ** (abs Y) = Base**(Exp_High + Exp_Mid + Exp_Low) * Factor Base : Double := X; Exp_High : Double := Double'Floor (Abs_Y); Exp_Mid : Double; Exp_Low : Double; Exp_Int : Mantissa_Type; Factor : Double := 1.0; begin -- Select algorithm for calculating Pow: -- integer cases fall through if Exp_High >= 2.0**Double'Machine_Mantissa then -- In case of Y that is IEEE infinity, just raise constraint error if Exp_High > Double'Safe_Last then raise Constraint_Error; end if; -- Large values of Y are even integers and will stay integer -- after division by two. loop -- Exp_Mid and Exp_Low are zero, so -- X**(abs Y) = Base ** Exp_High = (Base**2) ** (Exp_High / 2) Exp_High := Exp_High / 2.0; Base := Base * Base; exit when Exp_High < 2.0**Double'Machine_Mantissa; end loop; elsif Exp_High /= Abs_Y then Exp_Low := Abs_Y - Exp_High; Factor := 1.0; if Exp_Low /= 0.0 then -- Exp_Low now is in interval (0.0, 1.0) -- Exp_Mid := Double'Floor (Exp_Low * 4.0) / 4.0; Exp_Mid := 0.0; Exp_Low := Exp_Low - Exp_Mid; if Exp_Low >= 0.5 then Factor := Sqrt (X); Exp_Low := Exp_Low - 0.5; -- exact if Exp_Low >= 0.25 then Factor := Factor * Sqrt (Factor); Exp_Low := Exp_Low - 0.25; -- exact end if; elsif Exp_Low >= 0.25 then Factor := Sqrt (Sqrt (X)); Exp_Low := Exp_Low - 0.25; -- exact end if; -- Exp_Low now is in interval (0.0, 0.25) -- This means it is safe to call Logarithmic_Pow -- for the remaining part. Factor := Factor * Logarithmic_Pow (X, Exp_Low); end if; elsif X = 0.0 then return 0.0; end if; -- Exp_High is non-zero integer smaller than 2**Double'Machine_Mantissa Exp_Int := Mantissa_Type (Exp_High); -- Standard way for processing integer powers > 0 while Exp_Int > 1 loop if (Exp_Int and 1) = 1 then -- Base**Y = Base**(Exp_Int - 1) * Exp_Int for Exp_Int > 0 Factor := Factor * Base; end if; -- Exp_Int is even and Exp_Int > 0, so -- Base**Y = (Base**2)**(Exp_Int / 2) Base := Base * Base; Exp_Int := Exp_Int / 2; end loop; -- Exp_Int = 1 or Exp_Int = 0 if Exp_Int = 1 then Factor := Base * Factor; end if; if Negative_Y then Factor := 1.0 / Factor; end if; return Factor; end Pow; --------- -- Sin -- --------- function Sin (X : Double) return Double is Reduced_X : Double := X; Result : Double; Status : FPU_Status_Word; begin loop Asm (Template => "fsin " & NL & "xorl %%eax, %%eax " & NL & "fnstsw %%ax ", Outputs => (Double'Asm_Output ("=t", Result), FPU_Status_Word'Asm_Output ("=a", Status)), Inputs => Double'Asm_Input ("0", Reduced_X)); exit when not Status.C2; -- Original argument was not in range and the result -- is the unmodified argument. Reduced_X := Reduce (Result); end loop; return Result; end Sin; --------- -- Tan -- --------- function Tan (X : Double) return Double is Reduced_X : Double := X; Result : Double; Status : FPU_Status_Word; begin loop Asm (Template => "fptan " & NL & "xorl %%eax, %%eax " & NL & "fnstsw %%ax " & NL & "ffree %%st(0) " & NL & "fincstp ", Outputs => (Double'Asm_Output ("=t", Result), FPU_Status_Word'Asm_Output ("=a", Status)), Inputs => Double'Asm_Input ("0", Reduced_X)); exit when not Status.C2; -- Original argument was not in range and the result -- is the unmodified argument. Reduced_X := Reduce (Result); end loop; return Result; end Tan; ---------- -- Sinh -- ---------- function Sinh (X : Double) return Double is begin -- Mathematically Sinh (x) is defined to be (Exp (X) - Exp (-X)) / 2.0 if abs X < 25.0 then return (Exp (X) - Exp (-X)) / 2.0; else return Exp (X) / 2.0; end if; end Sinh; ---------- -- Cosh -- ---------- function Cosh (X : Double) return Double is begin -- Mathematically Cosh (X) is defined to be (Exp (X) + Exp (-X)) / 2.0 if abs X < 22.0 then return (Exp (X) + Exp (-X)) / 2.0; else return Exp (X) / 2.0; end if; end Cosh; ---------- -- Tanh -- ---------- function Tanh (X : Double) return Double is begin -- Return the Hyperbolic Tangent of x -- -- x -x -- e - e Sinh (X) -- Tanh (X) is defined to be ----------- = -------- -- x -x Cosh (X) -- e + e if abs X > 23.0 then return Double'Copy_Sign (1.0, X); end if; return 1.0 / (1.0 + Exp (-2.0 * X)) - 1.0 / (1.0 + Exp (2.0 * X)); end Tanh; end Ada.Numerics.Aux;