------------------------------------------------------------------------------
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------------------------------------------------------------------------------
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-- --
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-- --
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-- GNAT RUN-TIME COMPONENTS --
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-- GNAT RUN-TIME COMPONENTS --
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-- --
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-- --
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-- A D A . N U M E R I C S . A U X --
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-- A D A . N U M E R I C S . A U X --
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-- --
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-- --
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-- B o d y --
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-- B o d y --
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-- (Apple OS X Version) --
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-- (Apple OS X Version) --
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-- --
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-- --
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-- Copyright (C) 1998-2009, Free Software Foundation, Inc. --
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-- Copyright (C) 1998-2009, Free Software Foundation, Inc. --
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-- --
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-- --
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-- GNAT is free software; you can redistribute it and/or modify it under --
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-- GNAT is free software; you can redistribute it and/or modify it under --
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-- terms of the GNU General Public License as published by the Free Soft- --
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-- terms of the GNU General Public License as published by the Free Soft- --
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-- ware Foundation; either version 3, or (at your option) any later ver- --
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-- ware Foundation; either version 3, or (at your option) any later ver- --
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-- sion. GNAT is distributed in the hope that it will be useful, but WITH- --
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-- sion. GNAT is distributed in the hope that it will be useful, but WITH- --
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-- OUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY --
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-- OUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY --
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-- or FITNESS FOR A PARTICULAR PURPOSE. --
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-- or FITNESS FOR A PARTICULAR PURPOSE. --
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-- --
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-- --
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-- As a special exception under Section 7 of GPL version 3, you are granted --
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-- As a special exception under Section 7 of GPL version 3, you are granted --
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-- additional permissions described in the GCC Runtime Library Exception, --
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-- additional permissions described in the GCC Runtime Library Exception, --
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-- version 3.1, as published by the Free Software Foundation. --
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-- version 3.1, as published by the Free Software Foundation. --
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-- --
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-- --
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-- You should have received a copy of the GNU General Public License and --
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-- You should have received a copy of the GNU General Public License and --
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-- a copy of the GCC Runtime Library Exception along with this program; --
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-- a copy of the GCC Runtime Library Exception along with this program; --
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-- see the files COPYING3 and COPYING.RUNTIME respectively. If not, see --
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-- see the files COPYING3 and COPYING.RUNTIME respectively. If not, see --
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-- <http://www.gnu.org/licenses/>. --
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-- <http://www.gnu.org/licenses/>. --
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-- --
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-- --
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-- GNAT was originally developed by the GNAT team at New York University. --
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-- GNAT was originally developed by the GNAT team at New York University. --
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-- Extensive contributions were provided by Ada Core Technologies Inc. --
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-- Extensive contributions were provided by Ada Core Technologies Inc. --
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-- --
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-- --
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------------------------------------------------------------------------------
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------------------------------------------------------------------------------
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-- File a-numaux.adb <- a-numaux-darwin.adb
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-- File a-numaux.adb <- a-numaux-darwin.adb
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package body Ada.Numerics.Aux is
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package body Ada.Numerics.Aux is
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-----------------------
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-----------------------
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-- Local subprograms --
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-- Local subprograms --
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-----------------------
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-----------------------
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procedure Reduce (X : in out Double; Q : out Natural);
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procedure Reduce (X : in out Double; Q : out Natural);
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-- Implements reduction of X by Pi/2. Q is the quadrant of the final
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-- Implements reduction of X by Pi/2. Q is the quadrant of the final
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-- result in the range 0 .. 3. The absolute value of X is at most Pi/4.
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-- result in the range 0 .. 3. The absolute value of X is at most Pi/4.
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-- The following three functions implement Chebishev approximations
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-- The following three functions implement Chebishev approximations
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-- of the trigonometric functions in their reduced domain.
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-- of the trigonometric functions in their reduced domain.
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-- These approximations have been computed using Maple.
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-- These approximations have been computed using Maple.
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function Sine_Approx (X : Double) return Double;
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function Sine_Approx (X : Double) return Double;
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function Cosine_Approx (X : Double) return Double;
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function Cosine_Approx (X : Double) return Double;
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pragma Inline (Reduce);
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pragma Inline (Reduce);
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pragma Inline (Sine_Approx);
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pragma Inline (Sine_Approx);
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pragma Inline (Cosine_Approx);
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pragma Inline (Cosine_Approx);
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function Cosine_Approx (X : Double) return Double is
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function Cosine_Approx (X : Double) return Double is
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XX : constant Double := X * X;
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XX : constant Double := X * X;
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begin
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begin
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return (((((16#8.DC57FBD05F640#E-08 * XX
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return (((((16#8.DC57FBD05F640#E-08 * XX
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- 16#4.9F7D00BF25D80#E-06) * XX
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- 16#4.9F7D00BF25D80#E-06) * XX
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+ 16#1.A019F7FDEFCC2#E-04) * XX
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+ 16#1.A019F7FDEFCC2#E-04) * XX
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- 16#5.B05B058F18B20#E-03) * XX
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- 16#5.B05B058F18B20#E-03) * XX
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+ 16#A.AAAAAAAA73FA8#E-02) * XX
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+ 16#A.AAAAAAAA73FA8#E-02) * XX
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- 16#7.FFFFFFFFFFDE4#E-01) * XX
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- 16#7.FFFFFFFFFFDE4#E-01) * XX
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- 16#3.655E64869ECCE#E-14 + 1.0;
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- 16#3.655E64869ECCE#E-14 + 1.0;
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end Cosine_Approx;
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end Cosine_Approx;
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function Sine_Approx (X : Double) return Double is
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function Sine_Approx (X : Double) return Double is
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XX : constant Double := X * X;
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XX : constant Double := X * X;
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begin
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begin
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return (((((16#A.EA2D4ABE41808#E-09 * XX
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return (((((16#A.EA2D4ABE41808#E-09 * XX
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- 16#6.B974C10F9D078#E-07) * XX
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- 16#6.B974C10F9D078#E-07) * XX
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+ 16#2.E3BC673425B0E#E-05) * XX
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+ 16#2.E3BC673425B0E#E-05) * XX
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- 16#D.00D00CCA7AF00#E-04) * XX
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- 16#D.00D00CCA7AF00#E-04) * XX
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+ 16#2.222222221B190#E-02) * XX
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+ 16#2.222222221B190#E-02) * XX
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- 16#2.AAAAAAAAAAA44#E-01) * (XX * X) + X;
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- 16#2.AAAAAAAAAAA44#E-01) * (XX * X) + X;
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end Sine_Approx;
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end Sine_Approx;
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------------
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------------
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-- Reduce --
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-- Reduce --
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------------
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------------
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procedure Reduce (X : in out Double; Q : out Natural) is
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procedure Reduce (X : in out Double; Q : out Natural) is
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Half_Pi : constant := Pi / 2.0;
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Half_Pi : constant := Pi / 2.0;
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Two_Over_Pi : constant := 2.0 / Pi;
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Two_Over_Pi : constant := 2.0 / Pi;
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HM : constant := Integer'Min (Double'Machine_Mantissa / 2, Natural'Size);
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HM : constant := Integer'Min (Double'Machine_Mantissa / 2, Natural'Size);
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M : constant Double := 0.5 + 2.0**(1 - HM); -- Splitting constant
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M : constant Double := 0.5 + 2.0**(1 - HM); -- Splitting constant
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P1 : constant Double := Double'Leading_Part (Half_Pi, HM);
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P1 : constant Double := Double'Leading_Part (Half_Pi, HM);
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P2 : constant Double := Double'Leading_Part (Half_Pi - P1, HM);
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P2 : constant Double := Double'Leading_Part (Half_Pi - P1, HM);
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P3 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2, HM);
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P3 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2, HM);
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P4 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2 - P3, HM);
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P4 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2 - P3, HM);
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P5 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2 - P3
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P5 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2 - P3
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- P4, HM);
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- P4, HM);
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P6 : constant Double := Double'Model (Half_Pi - P1 - P2 - P3 - P4 - P5);
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P6 : constant Double := Double'Model (Half_Pi - P1 - P2 - P3 - P4 - P5);
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K : Double;
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K : Double;
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begin
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begin
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-- For X < 2.0**HM, all products below are computed exactly.
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-- For X < 2.0**HM, all products below are computed exactly.
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-- Due to cancellation effects all subtractions are exact as well.
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-- Due to cancellation effects all subtractions are exact as well.
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-- As no double extended floating-point number has more than 75
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-- As no double extended floating-point number has more than 75
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-- zeros after the binary point, the result will be the correctly
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-- zeros after the binary point, the result will be the correctly
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-- rounded result of X - K * (Pi / 2.0).
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-- rounded result of X - K * (Pi / 2.0).
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K := X * Two_Over_Pi;
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K := X * Two_Over_Pi;
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while abs K >= 2.0 ** HM loop
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while abs K >= 2.0 ** HM loop
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K := K * M - (K * M - K);
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K := K * M - (K * M - K);
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X :=
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X :=
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(((((X - K * P1) - K * P2) - K * P3) - K * P4) - K * P5) - K * P6;
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(((((X - K * P1) - K * P2) - K * P3) - K * P4) - K * P5) - K * P6;
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K := X * Two_Over_Pi;
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K := X * Two_Over_Pi;
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end loop;
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end loop;
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-- If K is not a number (because X was not finite) raise exception
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-- If K is not a number (because X was not finite) raise exception
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if K /= K then
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if K /= K then
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raise Constraint_Error;
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raise Constraint_Error;
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end if;
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end if;
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K := Double'Rounding (K);
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K := Double'Rounding (K);
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Q := Integer (K) mod 4;
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Q := Integer (K) mod 4;
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X := (((((X - K * P1) - K * P2) - K * P3)
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X := (((((X - K * P1) - K * P2) - K * P3)
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- K * P4) - K * P5) - K * P6;
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- K * P4) - K * P5) - K * P6;
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end Reduce;
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end Reduce;
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---------
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---------
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-- Cos --
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-- Cos --
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---------
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---------
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function Cos (X : Double) return Double is
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function Cos (X : Double) return Double is
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Reduced_X : Double := abs X;
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Reduced_X : Double := abs X;
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Quadrant : Natural range 0 .. 3;
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Quadrant : Natural range 0 .. 3;
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begin
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begin
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if Reduced_X > Pi / 4.0 then
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if Reduced_X > Pi / 4.0 then
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Reduce (Reduced_X, Quadrant);
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Reduce (Reduced_X, Quadrant);
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case Quadrant is
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case Quadrant is
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when 0 =>
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when 0 =>
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return Cosine_Approx (Reduced_X);
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return Cosine_Approx (Reduced_X);
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when 1 =>
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when 1 =>
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return Sine_Approx (-Reduced_X);
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return Sine_Approx (-Reduced_X);
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when 2 =>
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when 2 =>
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return -Cosine_Approx (Reduced_X);
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return -Cosine_Approx (Reduced_X);
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when 3 =>
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when 3 =>
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return Sine_Approx (Reduced_X);
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return Sine_Approx (Reduced_X);
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end case;
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end case;
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end if;
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end if;
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return Cosine_Approx (Reduced_X);
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return Cosine_Approx (Reduced_X);
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end Cos;
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end Cos;
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---------
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---------
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-- Sin --
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-- Sin --
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---------
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---------
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function Sin (X : Double) return Double is
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function Sin (X : Double) return Double is
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Reduced_X : Double := X;
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Reduced_X : Double := X;
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Quadrant : Natural range 0 .. 3;
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Quadrant : Natural range 0 .. 3;
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begin
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begin
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if abs X > Pi / 4.0 then
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if abs X > Pi / 4.0 then
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Reduce (Reduced_X, Quadrant);
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Reduce (Reduced_X, Quadrant);
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case Quadrant is
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case Quadrant is
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when 0 =>
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when 0 =>
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return Sine_Approx (Reduced_X);
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return Sine_Approx (Reduced_X);
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when 1 =>
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when 1 =>
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return Cosine_Approx (Reduced_X);
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return Cosine_Approx (Reduced_X);
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when 2 =>
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when 2 =>
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return Sine_Approx (-Reduced_X);
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return Sine_Approx (-Reduced_X);
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when 3 =>
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when 3 =>
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return -Cosine_Approx (Reduced_X);
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return -Cosine_Approx (Reduced_X);
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end case;
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end case;
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end if;
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end if;
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return Sine_Approx (Reduced_X);
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return Sine_Approx (Reduced_X);
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end Sin;
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end Sin;
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end Ada.Numerics.Aux;
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end Ada.Numerics.Aux;
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