| ------------------------------------------------------------------------------ |
| -- -- |
| -- GNAT RUN-TIME COMPONENTS -- |
| -- -- |
| -- ADA.NUMERICS.BIG_NUMBERS.BIG_REALS -- |
| -- -- |
| -- B o d y -- |
| -- -- |
| -- Copyright (C) 2019-2023, 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. -- |
| -- -- |
| ------------------------------------------------------------------------------ |
| |
| with System.Unsigned_Types; use System.Unsigned_Types; |
| |
| package body Ada.Numerics.Big_Numbers.Big_Reals is |
| |
| use Big_Integers; |
| |
| procedure Normalize (Arg : in out Big_Real); |
| -- Normalize Arg by ensuring that Arg.Den is always positive and that |
| -- Arg.Num and Arg.Den always have a GCD of 1. |
| |
| -------------- |
| -- Is_Valid -- |
| -------------- |
| |
| function Is_Valid (Arg : Big_Real) return Boolean is |
| (Is_Valid (Arg.Num) and Is_Valid (Arg.Den)); |
| |
| --------- |
| -- "/" -- |
| --------- |
| |
| function "/" (Num, Den : Valid_Big_Integer) return Valid_Big_Real is |
| Result : Big_Real; |
| begin |
| if Den = To_Big_Integer (0) then |
| raise Constraint_Error with "divide by zero"; |
| end if; |
| |
| Result.Num := Num; |
| Result.Den := Den; |
| Normalize (Result); |
| return Result; |
| end "/"; |
| |
| --------------- |
| -- Numerator -- |
| --------------- |
| |
| function Numerator (Arg : Valid_Big_Real) return Valid_Big_Integer is |
| (Arg.Num); |
| |
| ----------------- |
| -- Denominator -- |
| ----------------- |
| |
| function Denominator (Arg : Valid_Big_Real) return Big_Positive is |
| (Arg.Den); |
| |
| --------- |
| -- "=" -- |
| --------- |
| |
| function "=" (L, R : Valid_Big_Real) return Boolean is |
| (L.Num = R.Num and then L.Den = R.Den); |
| |
| --------- |
| -- "<" -- |
| --------- |
| |
| function "<" (L, R : Valid_Big_Real) return Boolean is |
| (L.Num * R.Den < R.Num * L.Den); |
| -- The denominator is guaranteed to be positive since Normalized is |
| -- always called when constructing a Valid_Big_Real |
| |
| ---------- |
| -- "<=" -- |
| ---------- |
| |
| function "<=" (L, R : Valid_Big_Real) return Boolean is (not (R < L)); |
| |
| --------- |
| -- ">" -- |
| --------- |
| |
| function ">" (L, R : Valid_Big_Real) return Boolean is (R < L); |
| |
| ---------- |
| -- ">=" -- |
| ---------- |
| |
| function ">=" (L, R : Valid_Big_Real) return Boolean is (not (L < R)); |
| |
| ----------------------- |
| -- Float_Conversions -- |
| ----------------------- |
| |
| package body Float_Conversions is |
| |
| package Conv is new |
| Big_Integers.Unsigned_Conversions (Long_Long_Unsigned); |
| |
| ----------------- |
| -- To_Big_Real -- |
| ----------------- |
| |
| -- We get the fractional representation of the floating-point number by |
| -- multiplying Num'Fraction by 2.0**M, with M the size of the mantissa, |
| -- which gives zero or a number in the range [2.0**(M-1)..2.0**M), which |
| -- means that it is an integer N of M bits. The floating-point number is |
| -- thus equal to N / 2**(M-E) where E is its Num'Exponent. |
| |
| function To_Big_Real (Arg : Num) return Valid_Big_Real is |
| |
| A : constant Num'Base := abs (Arg); |
| E : constant Integer := Num'Exponent (A); |
| F : constant Num'Base := Num'Fraction (A); |
| M : constant Natural := Num'Machine_Mantissa; |
| |
| N, D : Big_Integer; |
| |
| begin |
| pragma Assert (Num'Machine_Radix = 2); |
| -- This implementation does not handle radix 16 |
| |
| pragma Assert (M <= 64); |
| -- This implementation handles only 80-bit IEEE Extended or smaller |
| |
| N := Conv.To_Big_Integer (Long_Long_Unsigned (F * 2.0**M)); |
| |
| -- If E is smaller than M, the denominator is 2**(M-E) |
| |
| if E < M then |
| D := To_Big_Integer (2) ** (M - E); |
| |
| -- Or else, if E is larger than M, multiply the numerator by 2**(E-M) |
| |
| elsif E > M then |
| N := N * To_Big_Integer (2) ** (E - M); |
| D := To_Big_Integer (1); |
| |
| -- Otherwise E is equal to M and the result is just N |
| |
| else |
| D := To_Big_Integer (1); |
| end if; |
| |
| return (if Arg >= 0.0 then N / D else -N / D); |
| end To_Big_Real; |
| |
| ------------------- |
| -- From_Big_Real -- |
| ------------------- |
| |
| -- We get the (Frac, Exp) representation of the real number by finding |
| -- the exponent E such that it lies in the range [2.0**(E-1)..2.0**E), |
| -- multiplying the number by 2.0**(M-E) with M the size of the mantissa, |
| -- and converting the result to integer N in the range [2**(M-1)..2**M) |
| -- with rounding to nearest, ties to even, and finally call Num'Compose. |
| -- This does not apply to the zero, for which we return 0.0 early. |
| |
| function From_Big_Real (Arg : Big_Real) return Num is |
| |
| M : constant Natural := Num'Machine_Mantissa; |
| One : constant Big_Real := To_Real (1); |
| Two : constant Big_Real := To_Real (2); |
| Half : constant Big_Real := One / Two; |
| TwoI : constant Big_Integer := To_Big_Integer (2); |
| |
| function Log2_Estimate (V : Big_Real) return Natural; |
| -- Return an integer not larger than Log2 (V) for V >= 1.0 |
| |
| function Minus_Log2_Estimate (V : Big_Real) return Natural; |
| -- Return an integer not larger than -Log2 (V) for V < 1.0 |
| |
| ------------------- |
| -- Log2_Estimate -- |
| ------------------- |
| |
| function Log2_Estimate (V : Big_Real) return Natural is |
| Log : Natural := 1; |
| Pow : Big_Real := Two; |
| |
| begin |
| while V >= Pow loop |
| Pow := Pow * Pow; |
| Log := Log + Log; |
| end loop; |
| |
| return Log / 2; |
| end Log2_Estimate; |
| |
| ------------------------- |
| -- Minus_Log2_Estimate -- |
| ------------------------- |
| |
| function Minus_Log2_Estimate (V : Big_Real) return Natural is |
| Log : Natural := 1; |
| Pow : Big_Real := Half; |
| |
| begin |
| while V <= Pow loop |
| Pow := Pow * Pow; |
| Log := Log + Log; |
| end loop; |
| |
| return Log / 2; |
| end Minus_Log2_Estimate; |
| |
| -- Local variables |
| |
| V : Big_Real := abs (Arg); |
| E : Integer := 0; |
| L : Integer; |
| |
| A, B, Q, X : Big_Integer; |
| N : Long_Long_Unsigned; |
| R : Num'Base; |
| |
| begin |
| pragma Assert (Num'Machine_Radix = 2); |
| -- This implementation does not handle radix 16 |
| |
| pragma Assert (M <= 64); |
| -- This implementation handles only 80-bit IEEE Extended or smaller |
| |
| -- Protect from degenerate case |
| |
| if Numerator (V) = To_Big_Integer (0) then |
| return 0.0; |
| end if; |
| |
| -- Use a binary search to compute exponent E |
| |
| while V < Half loop |
| L := Minus_Log2_Estimate (V); |
| V := V * (Two ** L); |
| E := E - L; |
| end loop; |
| |
| -- The dissymetry with above is expected since we go below 2 |
| |
| while V >= One loop |
| L := Log2_Estimate (V) + 1; |
| V := V / (Two ** L); |
| E := E + L; |
| end loop; |
| |
| -- The multiplication by 2.0**(-E) has already been done in the loops |
| |
| V := V * To_Big_Real (TwoI ** M); |
| |
| -- Now go into the integer domain and divide |
| |
| A := Numerator (V); |
| B := Denominator (V); |
| |
| Q := A / B; |
| N := Conv.From_Big_Integer (Q); |
| |
| -- Round to nearest, ties to even, by comparing twice the remainder |
| |
| X := (A - Q * B) * TwoI; |
| |
| if X > B or else (X = B and then (N mod 2) = 1) then |
| N := N + 1; |
| |
| -- If the adjusted quotient overflows the mantissa, scale up |
| |
| if N = 2**M then |
| N := 1; |
| E := E + 1; |
| end if; |
| end if; |
| |
| R := Num'Compose (Num'Base (N), E); |
| |
| return (if Numerator (Arg) >= To_Big_Integer (0) then R else -R); |
| end From_Big_Real; |
| |
| end Float_Conversions; |
| |
| ----------------------- |
| -- Fixed_Conversions -- |
| ----------------------- |
| |
| package body Fixed_Conversions is |
| |
| package Float_Aux is new Float_Conversions (Long_Float); |
| |
| subtype LLLI is Long_Long_Long_Integer; |
| subtype LLLU is Long_Long_Long_Unsigned; |
| |
| Too_Large : constant Boolean := |
| Num'Small_Numerator > LLLU'Last |
| or else Num'Small_Denominator > LLLU'Last; |
| -- True if the Small is too large for Long_Long_Long_Unsigned, in which |
| -- case we convert to/from Long_Float as an intermediate step. |
| |
| package Conv_I is new Big_Integers.Signed_Conversions (LLLI); |
| package Conv_U is new Big_Integers.Unsigned_Conversions (LLLU); |
| |
| ----------------- |
| -- To_Big_Real -- |
| ----------------- |
| |
| -- We just compute V * N / D where V is the mantissa value of the fixed |
| -- point number, and N resp. D is the numerator resp. the denominator of |
| -- the Small of the fixed-point type. |
| |
| function To_Big_Real (Arg : Num) return Valid_Big_Real is |
| N, D, V : Big_Integer; |
| |
| begin |
| if Too_Large then |
| return Float_Aux.To_Big_Real (Long_Float (Arg)); |
| end if; |
| |
| N := Conv_U.To_Big_Integer (Num'Small_Numerator); |
| D := Conv_U.To_Big_Integer (Num'Small_Denominator); |
| V := Conv_I.To_Big_Integer (LLLI'Integer_Value (Arg)); |
| |
| return V * N / D; |
| end To_Big_Real; |
| |
| ------------------- |
| -- From_Big_Real -- |
| ------------------- |
| |
| -- We first compute A / B = Arg * D / N where N resp. D is the numerator |
| -- resp. the denominator of the Small of the fixed-point type. Then we |
| -- divide A by B and convert the result to the mantissa value. |
| |
| function From_Big_Real (Arg : Big_Real) return Num is |
| N, D, A, B, Q, X : Big_Integer; |
| |
| begin |
| if Too_Large then |
| return Num (Float_Aux.From_Big_Real (Arg)); |
| end if; |
| |
| N := Conv_U.To_Big_Integer (Num'Small_Numerator); |
| D := Conv_U.To_Big_Integer (Num'Small_Denominator); |
| A := Numerator (Arg) * D; |
| B := Denominator (Arg) * N; |
| |
| Q := A / B; |
| |
| -- Round to nearest, ties to away, by comparing twice the remainder |
| |
| X := (A - Q * B) * To_Big_Integer (2); |
| |
| if X >= B then |
| Q := Q + To_Big_Integer (1); |
| |
| elsif X <= -B then |
| Q := Q - To_Big_Integer (1); |
| end if; |
| |
| return Num'Fixed_Value (Conv_I.From_Big_Integer (Q)); |
| end From_Big_Real; |
| |
| end Fixed_Conversions; |
| |
| --------------- |
| -- To_String -- |
| --------------- |
| |
| function To_String |
| (Arg : Valid_Big_Real; |
| Fore : Field := 2; |
| Aft : Field := 3; |
| Exp : Field := 0) return String |
| is |
| Zero : constant Big_Integer := To_Big_Integer (0); |
| Ten : constant Big_Integer := To_Big_Integer (10); |
| |
| function Leading_Padding |
| (Str : String; |
| Min_Length : Field; |
| Char : Character := ' ') return String; |
| -- Return padding of Char concatenated with Str so that the resulting |
| -- string is at least Min_Length long. |
| |
| function Trailing_Padding |
| (Str : String; |
| Length : Field; |
| Char : Character := '0') return String; |
| -- Return Str with trailing Char removed, and if needed either |
| -- truncated or concatenated with padding of Char so that the resulting |
| -- string is Length long. |
| |
| function Image (N : Natural) return String; |
| -- Return image of N, with no leading space. |
| |
| function Numerator_Image |
| (Num : Big_Integer; |
| After : Natural) return String; |
| -- Return image of Num as a float value with After digits after the "." |
| -- and taking Fore, Aft, Exp into account. |
| |
| ----------- |
| -- Image -- |
| ----------- |
| |
| function Image (N : Natural) return String is |
| S : constant String := Natural'Image (N); |
| begin |
| return S (2 .. S'Last); |
| end Image; |
| |
| --------------------- |
| -- Leading_Padding -- |
| --------------------- |
| |
| function Leading_Padding |
| (Str : String; |
| Min_Length : Field; |
| Char : Character := ' ') return String is |
| begin |
| if Str = "" then |
| return Leading_Padding ("0", Min_Length, Char); |
| else |
| return [1 .. Integer'Max (Integer (Min_Length) - Str'Length, 0) |
| => Char] & Str; |
| end if; |
| end Leading_Padding; |
| |
| ---------------------- |
| -- Trailing_Padding -- |
| ---------------------- |
| |
| function Trailing_Padding |
| (Str : String; |
| Length : Field; |
| Char : Character := '0') return String is |
| begin |
| if Str'Length > 0 and then Str (Str'Last) = Char then |
| for J in reverse Str'Range loop |
| if Str (J) /= '0' then |
| return Trailing_Padding |
| (Str (Str'First .. J), Length, Char); |
| end if; |
| end loop; |
| end if; |
| |
| if Str'Length >= Length then |
| return Str (Str'First .. Str'First + Length - 1); |
| else |
| return Str & |
| [1 .. Integer'Max (Integer (Length) - Str'Length, 0) |
| => Char]; |
| end if; |
| end Trailing_Padding; |
| |
| --------------------- |
| -- Numerator_Image -- |
| --------------------- |
| |
| function Numerator_Image |
| (Num : Big_Integer; |
| After : Natural) return String |
| is |
| Tmp : constant String := To_String (Num); |
| Str : constant String (1 .. Tmp'Last - 1) := Tmp (2 .. Tmp'Last); |
| Index : Integer; |
| |
| begin |
| if After = 0 then |
| return Leading_Padding (Str, Fore) & "." |
| & Trailing_Padding ("0", Aft); |
| else |
| Index := Str'Last - After; |
| |
| if Index < 0 then |
| return Leading_Padding ("0", Fore) |
| & "." |
| & Trailing_Padding ([1 .. -Index => '0'] & Str, Aft) |
| & (if Exp = 0 then "" else "E+" & Image (Natural (Exp))); |
| else |
| return Leading_Padding (Str (Str'First .. Index), Fore) |
| & "." |
| & Trailing_Padding (Str (Index + 1 .. Str'Last), Aft) |
| & (if Exp = 0 then "" else "E+" & Image (Natural (Exp))); |
| end if; |
| end if; |
| end Numerator_Image; |
| |
| begin |
| if Arg.Num < Zero then |
| declare |
| Str : String := To_String (-Arg, Fore, Aft, Exp); |
| begin |
| if Str (1) = ' ' then |
| for J in 1 .. Str'Last - 1 loop |
| if Str (J + 1) /= ' ' then |
| Str (J) := '-'; |
| exit; |
| end if; |
| end loop; |
| |
| return Str; |
| else |
| return '-' & Str; |
| end if; |
| end; |
| else |
| -- Compute Num * 10^Aft so that we get Aft significant digits |
| -- in the integer part (rounded) to display. |
| |
| return Numerator_Image |
| ((Arg.Num * Ten ** Aft) / Arg.Den, After => Exp + Aft); |
| end if; |
| end To_String; |
| |
| ----------------- |
| -- From_String -- |
| ----------------- |
| |
| function From_String (Arg : String) return Valid_Big_Real is |
| Ten : constant Big_Integer := To_Big_Integer (10); |
| Frac : Big_Integer; |
| Exp : Integer := 0; |
| Pow : Natural := 0; |
| Index : Natural := 0; |
| Last : Natural := Arg'Last; |
| |
| begin |
| for J in reverse Arg'Range loop |
| if Arg (J) in 'e' | 'E' then |
| if Last /= Arg'Last then |
| raise Constraint_Error with "multiple exponents specified"; |
| end if; |
| |
| Last := J - 1; |
| Exp := Integer'Value (Arg (J + 1 .. Arg'Last)); |
| Pow := 0; |
| |
| elsif Arg (J) = '.' then |
| Index := J - 1; |
| exit; |
| elsif Arg (J) /= '_' then |
| Pow := Pow + 1; |
| end if; |
| end loop; |
| |
| if Index = 0 then |
| raise Constraint_Error with "invalid real value"; |
| end if; |
| |
| declare |
| Result : Big_Real; |
| begin |
| Result.Den := Ten ** Pow; |
| Result.Num := From_String (Arg (Arg'First .. Index)) * Result.Den; |
| Frac := From_String (Arg (Index + 2 .. Last)); |
| |
| if Result.Num < To_Big_Integer (0) then |
| Result.Num := Result.Num - Frac; |
| else |
| Result.Num := Result.Num + Frac; |
| end if; |
| |
| if Exp > 0 then |
| Result.Num := Result.Num * Ten ** Exp; |
| elsif Exp < 0 then |
| Result.Den := Result.Den * Ten ** (-Exp); |
| end if; |
| |
| Normalize (Result); |
| return Result; |
| end; |
| end From_String; |
| |
| -------------------------- |
| -- From_Quotient_String -- |
| -------------------------- |
| |
| function From_Quotient_String (Arg : String) return Valid_Big_Real is |
| Index : Natural := 0; |
| begin |
| for J in Arg'First + 1 .. Arg'Last - 1 loop |
| if Arg (J) = '/' then |
| Index := J; |
| exit; |
| end if; |
| end loop; |
| |
| if Index = 0 then |
| raise Constraint_Error with "no quotient found"; |
| end if; |
| |
| return Big_Integers.From_String (Arg (Arg'First .. Index - 1)) / |
| Big_Integers.From_String (Arg (Index + 1 .. Arg'Last)); |
| end From_Quotient_String; |
| |
| --------------- |
| -- Put_Image -- |
| --------------- |
| |
| procedure Put_Image (S : in out Root_Buffer_Type'Class; V : Big_Real) is |
| -- This is implemented in terms of To_String. It might be more elegant |
| -- and more efficient to do it the other way around, but this is the |
| -- most expedient implementation for now. |
| begin |
| Strings.Text_Buffers.Put_UTF_8 (S, To_String (V)); |
| end Put_Image; |
| |
| --------- |
| -- "+" -- |
| --------- |
| |
| function "+" (L : Valid_Big_Real) return Valid_Big_Real is |
| Result : Big_Real; |
| begin |
| Result.Num := L.Num; |
| Result.Den := L.Den; |
| return Result; |
| end "+"; |
| |
| --------- |
| -- "-" -- |
| --------- |
| |
| function "-" (L : Valid_Big_Real) return Valid_Big_Real is |
| (Num => -L.Num, Den => L.Den); |
| |
| ----------- |
| -- "abs" -- |
| ----------- |
| |
| function "abs" (L : Valid_Big_Real) return Valid_Big_Real is |
| (Num => abs L.Num, Den => L.Den); |
| |
| --------- |
| -- "+" -- |
| --------- |
| |
| function "+" (L, R : Valid_Big_Real) return Valid_Big_Real is |
| Result : Big_Real; |
| begin |
| Result.Num := L.Num * R.Den + R.Num * L.Den; |
| Result.Den := L.Den * R.Den; |
| Normalize (Result); |
| return Result; |
| end "+"; |
| |
| --------- |
| -- "-" -- |
| --------- |
| |
| function "-" (L, R : Valid_Big_Real) return Valid_Big_Real is |
| Result : Big_Real; |
| begin |
| Result.Num := L.Num * R.Den - R.Num * L.Den; |
| Result.Den := L.Den * R.Den; |
| Normalize (Result); |
| return Result; |
| end "-"; |
| |
| --------- |
| -- "*" -- |
| --------- |
| |
| function "*" (L, R : Valid_Big_Real) return Valid_Big_Real is |
| Result : Big_Real; |
| begin |
| Result.Num := L.Num * R.Num; |
| Result.Den := L.Den * R.Den; |
| Normalize (Result); |
| return Result; |
| end "*"; |
| |
| --------- |
| -- "/" -- |
| --------- |
| |
| function "/" (L, R : Valid_Big_Real) return Valid_Big_Real is |
| Result : Big_Real; |
| begin |
| Result.Num := L.Num * R.Den; |
| Result.Den := L.Den * R.Num; |
| Normalize (Result); |
| return Result; |
| end "/"; |
| |
| ---------- |
| -- "**" -- |
| ---------- |
| |
| function "**" (L : Valid_Big_Real; R : Integer) return Valid_Big_Real is |
| Result : Big_Real; |
| begin |
| if R = 0 then |
| Result.Num := To_Big_Integer (1); |
| Result.Den := To_Big_Integer (1); |
| else |
| if R < 0 then |
| Result.Num := L.Den ** (-R); |
| Result.Den := L.Num ** (-R); |
| else |
| Result.Num := L.Num ** R; |
| Result.Den := L.Den ** R; |
| end if; |
| |
| Normalize (Result); |
| end if; |
| |
| return Result; |
| end "**"; |
| |
| --------- |
| -- Min -- |
| --------- |
| |
| function Min (L, R : Valid_Big_Real) return Valid_Big_Real is |
| (if L < R then L else R); |
| |
| --------- |
| -- Max -- |
| --------- |
| |
| function Max (L, R : Valid_Big_Real) return Valid_Big_Real is |
| (if L > R then L else R); |
| |
| --------------- |
| -- Normalize -- |
| --------------- |
| |
| procedure Normalize (Arg : in out Big_Real) is |
| Zero : constant Big_Integer := To_Big_Integer (0); |
| begin |
| if Arg.Den < Zero then |
| Arg.Num := -Arg.Num; |
| Arg.Den := -Arg.Den; |
| end if; |
| |
| if Arg.Num = Zero then |
| Arg.Den := To_Big_Integer (1); |
| else |
| declare |
| GCD : constant Big_Integer := |
| Greatest_Common_Divisor (Arg.Num, Arg.Den); |
| begin |
| Arg.Num := Arg.Num / GCD; |
| Arg.Den := Arg.Den / GCD; |
| end; |
| end if; |
| end Normalize; |
| |
| end Ada.Numerics.Big_Numbers.Big_Reals; |
