1 ------------------------------------------------------------------------------
3 -- GNAT RUNTIME COMPONENTS --
5 -- ADA.NUMERICS.GENERIC_ELEMENTARY_FUNCTIONS --
9 -- Copyright (C) 1992-2001, Free Software Foundation, Inc. --
11 -- GNAT is free software; you can redistribute it and/or modify it under --
12 -- terms of the GNU General Public License as published by the Free Soft- --
13 -- ware Foundation; either version 2, or (at your option) any later ver- --
14 -- sion. GNAT is distributed in the hope that it will be useful, but WITH- --
15 -- OUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY --
16 -- or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License --
17 -- for more details. You should have received a copy of the GNU General --
18 -- Public License distributed with GNAT; see file COPYING. If not, write --
19 -- to the Free Software Foundation, 59 Temple Place - Suite 330, Boston, --
20 -- MA 02111-1307, USA. --
22 -- As a special exception, if other files instantiate generics from this --
23 -- unit, or you link this unit with other files to produce an executable, --
24 -- this unit does not by itself cause the resulting executable to be --
25 -- covered by the GNU General Public License. This exception does not --
26 -- however invalidate any other reasons why the executable file might be --
27 -- covered by the GNU Public License. --
29 -- GNAT was originally developed by the GNAT team at New York University. --
30 -- Extensive contributions were provided by Ada Core Technologies Inc. --
32 ------------------------------------------------------------------------------
34 -- This body is specifically for using an Ada interface to C math.h to get
35 -- the computation engine. Many special cases are handled locally to avoid
36 -- unnecessary calls. This is not a "strict" implementation, but takes full
37 -- advantage of the C functions, e.g. in providing interface to hardware
38 -- provided versions of the elementary functions.
40 -- Uses functions sqrt, exp, log, pow, sin, asin, cos, acos, tan, atan,
41 -- sinh, cosh, tanh from C library via math.h
43 with Ada.Numerics.Aux;
45 package body Ada.Numerics.Generic_Elementary_Functions is
47 use type Ada.Numerics.Aux.Double;
49 Sqrt_Two : constant := 1.41421_35623_73095_04880_16887_24209_69807_85696;
50 Log_Two : constant := 0.69314_71805_59945_30941_72321_21458_17656_80755;
51 Half_Log_Two : constant := Log_Two / 2;
53 subtype T is Float_Type'Base;
54 subtype Double is Aux.Double;
56 Two_Pi : constant T := 2.0 * Pi;
57 Half_Pi : constant T := Pi / 2.0;
58 Fourth_Pi : constant T := Pi / 4.0;
60 Epsilon : constant T := 2.0 ** (1 - T'Model_Mantissa);
61 IEpsilon : constant T := 2.0 ** (T'Model_Mantissa - 1);
62 Log_Epsilon : constant T := T (1 - T'Model_Mantissa) * Log_Two;
63 Half_Log_Epsilon : constant T := T (1 - T'Model_Mantissa) * Half_Log_Two;
64 Log_Inverse_Epsilon : constant T := T (T'Model_Mantissa - 1) * Log_Two;
65 Sqrt_Epsilon : constant T := Sqrt_Two ** (1 - T'Model_Mantissa);
67 DEpsilon : constant Double := Double (Epsilon);
68 DIEpsilon : constant Double := Double (IEpsilon);
70 -----------------------
71 -- Local Subprograms --
72 -----------------------
74 function Exp_Strict (X : Float_Type'Base) return Float_Type'Base;
75 -- Cody/Waite routine, supposedly more precise than the library
76 -- version. Currently only needed for Sinh/Cosh on X86 with the largest
81 X : Float_Type'Base := 1.0)
82 return Float_Type'Base;
83 -- Common code for arc tangent after cyele reduction
89 function "**" (Left, Right : Float_Type'Base) return Float_Type'Base is
90 A_Right : Float_Type'Base;
92 Result : Float_Type'Base;
94 Rest : Float_Type'Base;
100 raise Argument_Error;
102 elsif Left < 0.0 then
103 raise Argument_Error;
105 elsif Right = 0.0 then
108 elsif Left = 0.0 then
110 raise Constraint_Error;
115 elsif Left = 1.0 then
118 elsif Right = 1.0 then
126 elsif Right = 0.5 then
130 A_Right := abs (Right);
132 -- If exponent is larger than one, compute integer exponen-
133 -- tiation if possible, and evaluate fractional part with
134 -- more precision. The relative error is now proportional
135 -- to the fractional part of the exponent only.
138 and then A_Right < Float_Type'Base (Integer'Last)
140 Int_Part := Integer (Float_Type'Base'Truncation (A_Right));
141 Result := Left ** Int_Part;
142 Rest := A_Right - Float_Type'Base (Int_Part);
144 -- Compute with two leading bits of the mantissa using
145 -- square roots. Bound to be better than logarithms, and
146 -- easily extended to greater precision.
150 Result := Result * R1;
154 Result := Result * Sqrt (R1);
158 elsif Rest >= 0.25 then
159 Result := Result * Sqrt (Sqrt (Left));
164 Float_Type'Base (Aux.Pow (Double (Left), Double (Rest)));
169 return (1.0 / Result);
173 Float_Type'Base (Aux.Pow (Double (Left), Double (Right)));
179 raise Constraint_Error;
190 function Arccos (X : Float_Type'Base) return Float_Type'Base is
191 Temp : Float_Type'Base;
195 raise Argument_Error;
197 elsif abs X < Sqrt_Epsilon then
207 Temp := Float_Type'Base (Aux.Acos (Double (X)));
218 function Arccos (X, Cycle : Float_Type'Base) return Float_Type'Base is
219 Temp : Float_Type'Base;
223 raise Argument_Error;
225 elsif abs X > 1.0 then
226 raise Argument_Error;
228 elsif abs X < Sqrt_Epsilon then
238 Temp := Arctan (Sqrt ((1.0 - X) * (1.0 + X)) / X, 1.0, Cycle);
241 Temp := Cycle / 2.0 + Temp;
251 function Arccosh (X : Float_Type'Base) return Float_Type'Base is
253 -- Return positive branch of Log (X - Sqrt (X * X - 1.0)), or
254 -- the proper approximation for X close to 1 or >> 1.
257 raise Argument_Error;
259 elsif X < 1.0 + Sqrt_Epsilon then
260 return Sqrt (2.0 * (X - 1.0));
262 elsif X > 1.0 / Sqrt_Epsilon then
263 return Log (X) + Log_Two;
266 return Log (X + Sqrt ((X - 1.0) * (X + 1.0)));
277 (X : Float_Type'Base;
278 Y : Float_Type'Base := 1.0)
279 return Float_Type'Base
282 -- Just reverse arguments
284 return Arctan (Y, X);
290 (X : Float_Type'Base;
291 Y : Float_Type'Base := 1.0;
292 Cycle : Float_Type'Base)
293 return Float_Type'Base
296 -- Just reverse arguments
298 return Arctan (Y, X, Cycle);
305 function Arccoth (X : Float_Type'Base) return Float_Type'Base is
308 return Arctanh (1.0 / X);
310 elsif abs X = 1.0 then
311 raise Constraint_Error;
313 elsif abs X < 1.0 then
314 raise Argument_Error;
317 -- 1.0 < abs X <= 2.0. One of X + 1.0 and X - 1.0 is exact, the
318 -- other has error 0 or Epsilon.
320 return 0.5 * (Log (abs (X + 1.0)) - Log (abs (X - 1.0)));
330 function Arcsin (X : Float_Type'Base) return Float_Type'Base is
333 raise Argument_Error;
335 elsif abs X < Sqrt_Epsilon then
345 return Float_Type'Base (Aux.Asin (Double (X)));
350 function Arcsin (X, Cycle : Float_Type'Base) return Float_Type'Base is
353 raise Argument_Error;
355 elsif abs X > 1.0 then
356 raise Argument_Error;
368 return Arctan (X / Sqrt ((1.0 - X) * (1.0 + X)), 1.0, Cycle);
375 function Arcsinh (X : Float_Type'Base) return Float_Type'Base is
377 if abs X < Sqrt_Epsilon then
380 elsif X > 1.0 / Sqrt_Epsilon then
381 return Log (X) + Log_Two;
383 elsif X < -1.0 / Sqrt_Epsilon then
384 return -(Log (-X) + Log_Two);
387 return -Log (abs X + Sqrt (X * X + 1.0));
390 return Log (X + Sqrt (X * X + 1.0));
401 (Y : Float_Type'Base;
402 X : Float_Type'Base := 1.0)
403 return Float_Type'Base
409 raise Argument_Error;
415 return Pi * Float_Type'Copy_Sign (1.0, Y);
426 return Local_Atan (Y, X);
433 (Y : Float_Type'Base;
434 X : Float_Type'Base := 1.0;
435 Cycle : Float_Type'Base)
436 return Float_Type'Base
440 raise Argument_Error;
445 raise Argument_Error;
451 return Cycle / 2.0 * Float_Type'Copy_Sign (1.0, Y);
462 return Local_Atan (Y, X) * Cycle / Two_Pi;
470 function Arctanh (X : Float_Type'Base) return Float_Type'Base is
471 A, B, D, A_Plus_1, A_From_1 : Float_Type'Base;
472 Mantissa : constant Integer := Float_Type'Base'Machine_Mantissa;
475 -- The naive formula:
477 -- Arctanh (X) := (1/2) * Log (1 + X) / (1 - X)
479 -- is not well-behaved numerically when X < 0.5 and when X is close
480 -- to one. The following is accurate but probably not optimal.
483 raise Constraint_Error;
485 elsif abs X >= 1.0 - 2.0 ** (-Mantissa) then
488 raise Argument_Error;
491 -- The one case that overflows if put through the method below:
492 -- abs X = 1.0 - Epsilon. In this case (1/2) log (2/Epsilon) is
493 -- accurate. This simplifies to:
495 return Float_Type'Copy_Sign (
496 Half_Log_Two * Float_Type'Base (Mantissa + 1), X);
499 -- elsif abs X <= 0.5 then
500 -- why is above line commented out ???
503 -- Use several piecewise linear approximations.
504 -- A is close to X, chosen so 1.0 + A, 1.0 - A, and X - A are exact.
505 -- The two scalings remove the low-order bits of X.
507 A := Float_Type'Base'Scaling (
508 Float_Type'Base (Long_Long_Integer
509 (Float_Type'Base'Scaling (X, Mantissa - 1))), 1 - Mantissa);
511 B := X - A; -- This is exact; abs B <= 2**(-Mantissa).
512 A_Plus_1 := 1.0 + A; -- This is exact.
513 A_From_1 := 1.0 - A; -- Ditto.
514 D := A_Plus_1 * A_From_1; -- 1 - A*A.
516 -- use one term of the series expansion:
517 -- f (x + e) = f(x) + e * f'(x) + ..
519 -- The derivative of Arctanh at A is 1/(1-A*A). Next term is
520 -- A*(B/D)**2 (if a quadratic approximation is ever needed).
522 return 0.5 * (Log (A_Plus_1) - Log (A_From_1)) + B / D;
525 -- return 0.5 * Log ((X + 1.0) / (1.0 - X));
526 -- why are above lines commented out ???
536 function Cos (X : Float_Type'Base) return Float_Type'Base is
541 elsif abs X < Sqrt_Epsilon then
546 return Float_Type'Base (Aux.Cos (Double (X)));
551 function Cos (X, Cycle : Float_Type'Base) return Float_Type'Base is
553 -- Just reuse the code for Sin. The potential small
554 -- loss of speed is negligible with proper (front-end) inlining.
556 return -Sin (abs X - Cycle * 0.25, Cycle);
563 function Cosh (X : Float_Type'Base) return Float_Type'Base is
564 Lnv : constant Float_Type'Base := 8#0.542714#;
565 V2minus1 : constant Float_Type'Base := 0.13830_27787_96019_02638E-4;
566 Y : Float_Type'Base := abs X;
570 if Y < Sqrt_Epsilon then
573 elsif Y > Log_Inverse_Epsilon then
574 Z := Exp_Strict (Y - Lnv);
575 return (Z + V2minus1 * Z);
579 return 0.5 * (Z + 1.0 / Z);
590 function Cot (X : Float_Type'Base) return Float_Type'Base is
593 raise Constraint_Error;
595 elsif abs X < Sqrt_Epsilon then
599 return 1.0 / Float_Type'Base (Aux.Tan (Double (X)));
604 function Cot (X, Cycle : Float_Type'Base) return Float_Type'Base is
609 raise Argument_Error;
612 T := Float_Type'Base'Remainder (X, Cycle);
614 if T = 0.0 or abs T = 0.5 * Cycle then
615 raise Constraint_Error;
617 elsif abs T < Sqrt_Epsilon then
620 elsif abs T = 0.25 * Cycle then
624 T := T / Cycle * Two_Pi;
625 return Cos (T) / Sin (T);
633 function Coth (X : Float_Type'Base) return Float_Type'Base is
636 raise Constraint_Error;
638 elsif X < Half_Log_Epsilon then
641 elsif X > -Half_Log_Epsilon then
644 elsif abs X < Sqrt_Epsilon then
648 return 1.0 / Float_Type'Base (Aux.Tanh (Double (X)));
655 function Exp (X : Float_Type'Base) return Float_Type'Base is
656 Result : Float_Type'Base;
663 Result := Float_Type'Base (Aux.Exp (Double (X)));
665 -- Deal with case of Exp returning IEEE infinity. If Machine_Overflows
666 -- is False, then we can just leave it as an infinity (and indeed we
667 -- prefer to do so). But if Machine_Overflows is True, then we have
668 -- to raise a Constraint_Error exception as required by the RM.
670 if Float_Type'Machine_Overflows and then not Result'Valid then
671 raise Constraint_Error;
681 function Exp_Strict (X : Float_Type'Base) return Float_Type'Base is
685 P0 : constant := 0.25000_00000_00000_00000;
686 P1 : constant := 0.75753_18015_94227_76666E-2;
687 P2 : constant := 0.31555_19276_56846_46356E-4;
689 Q0 : constant := 0.5;
690 Q1 : constant := 0.56817_30269_85512_21787E-1;
691 Q2 : constant := 0.63121_89437_43985_02557E-3;
692 Q3 : constant := 0.75104_02839_98700_46114E-6;
694 C1 : constant := 8#0.543#;
695 C2 : constant := -2.1219_44400_54690_58277E-4;
696 Le : constant := 1.4426_95040_88896_34074;
698 XN : Float_Type'Base;
699 P, Q, R : Float_Type'Base;
706 XN := Float_Type'Base'Rounding (X * Le);
707 G := (X - XN * C1) - XN * C2;
709 P := G * ((P2 * Z + P1) * Z + P0);
710 Q := ((Q3 * Z + Q2) * Z + Q1) * Z + Q0;
711 R := 0.5 + P / (Q - P);
713 R := Float_Type'Base'Scaling (R, Integer (XN) + 1);
715 -- Deal with case of Exp returning IEEE infinity. If Machine_Overflows
716 -- is False, then we can just leave it as an infinity (and indeed we
717 -- prefer to do so). But if Machine_Overflows is True, then we have
718 -- to raise a Constraint_Error exception as required by the RM.
720 if Float_Type'Machine_Overflows and then not R'Valid then
721 raise Constraint_Error;
733 (Y : Float_Type'Base;
734 X : Float_Type'Base := 1.0)
735 return Float_Type'Base
738 Raw_Atan : Float_Type'Base;
741 if abs Y > abs X then
747 if Z < Sqrt_Epsilon then
751 Raw_Atan := Pi / 4.0;
754 Raw_Atan := Float_Type'Base (Aux.Atan (Double (Z)));
757 if abs Y > abs X then
758 Raw_Atan := Half_Pi - Raw_Atan;
770 return Pi - Raw_Atan;
772 return -(Pi - Raw_Atan);
783 function Log (X : Float_Type'Base) return Float_Type'Base is
786 raise Argument_Error;
789 raise Constraint_Error;
795 return Float_Type'Base (Aux.Log (Double (X)));
800 function Log (X, Base : Float_Type'Base) return Float_Type'Base is
803 raise Argument_Error;
805 elsif Base <= 0.0 or else Base = 1.0 then
806 raise Argument_Error;
809 raise Constraint_Error;
815 return Float_Type'Base (Aux.Log (Double (X)) / Aux.Log (Double (Base)));
824 function Sin (X : Float_Type'Base) return Float_Type'Base is
826 if abs X < Sqrt_Epsilon then
830 return Float_Type'Base (Aux.Sin (Double (X)));
835 function Sin (X, Cycle : Float_Type'Base) return Float_Type'Base is
840 raise Argument_Error;
843 -- Is this test really needed on any machine ???
847 T := Float_Type'Base'Remainder (X, Cycle);
849 -- The following two reductions reduce the argument
850 -- to the interval [-0.25 * Cycle, 0.25 * Cycle].
851 -- This reduction is exact and is needed to prevent
852 -- inaccuracy that may result if the sinus function
853 -- a different (more accurate) value of Pi in its
854 -- reduction than is used in the multiplication with Two_Pi.
856 if abs T > 0.25 * Cycle then
857 T := 0.5 * Float_Type'Copy_Sign (Cycle, T) - T;
860 -- Could test for 12.0 * abs T = Cycle, and return
861 -- an exact value in those cases. It is not clear that
862 -- this is worth the extra test though.
864 return Float_Type'Base (Aux.Sin (Double (T / Cycle * Two_Pi)));
871 function Sinh (X : Float_Type'Base) return Float_Type'Base is
872 Lnv : constant Float_Type'Base := 8#0.542714#;
873 V2minus1 : constant Float_Type'Base := 0.13830_27787_96019_02638E-4;
874 Y : Float_Type'Base := abs X;
875 F : constant Float_Type'Base := Y * Y;
878 Float_Digits_1_6 : constant Boolean := Float_Type'Digits < 7;
881 if Y < Sqrt_Epsilon then
884 elsif Y > Log_Inverse_Epsilon then
885 Z := Exp_Strict (Y - Lnv);
886 Z := Z + V2minus1 * Z;
890 if Float_Digits_1_6 then
892 -- Use expansion provided by Cody and Waite, p. 226. Note that
893 -- leading term of the polynomial in Q is exactly 1.0.
896 P0 : constant := -0.71379_3159E+1;
897 P1 : constant := -0.19033_3399E+0;
898 Q0 : constant := -0.42827_7109E+2;
901 Z := Y + Y * F * (P1 * F + P0) / (F + Q0);
906 P0 : constant := -0.35181_28343_01771_17881E+6;
907 P1 : constant := -0.11563_52119_68517_68270E+5;
908 P2 : constant := -0.16375_79820_26307_51372E+3;
909 P3 : constant := -0.78966_12741_73570_99479E+0;
910 Q0 : constant := -0.21108_77005_81062_71242E+7;
911 Q1 : constant := 0.36162_72310_94218_36460E+5;
912 Q2 : constant := -0.27773_52311_96507_01667E+3;
915 Z := Y + Y * F * (((P3 * F + P2) * F + P1) * F + P0)
916 / (((F + Q2) * F + Q1) * F + Q0);
922 Z := 0.5 * (Z - 1.0 / Z);
936 function Sqrt (X : Float_Type'Base) return Float_Type'Base is
939 raise Argument_Error;
941 -- Special case Sqrt (0.0) to preserve possible minus sign per IEEE
948 return Float_Type'Base (Aux.Sqrt (Double (X)));
957 function Tan (X : Float_Type'Base) return Float_Type'Base is
959 if abs X < Sqrt_Epsilon then
962 elsif abs X = Pi / 2.0 then
963 raise Constraint_Error;
966 return Float_Type'Base (Aux.Tan (Double (X)));
971 function Tan (X, Cycle : Float_Type'Base) return Float_Type'Base is
976 raise Argument_Error;
982 T := Float_Type'Base'Remainder (X, Cycle);
984 if abs T = 0.25 * Cycle then
985 raise Constraint_Error;
987 elsif abs T = 0.5 * Cycle then
991 T := T / Cycle * Two_Pi;
992 return Sin (T) / Cos (T);
1001 function Tanh (X : Float_Type'Base) return Float_Type'Base is
1002 P0 : constant Float_Type'Base := -0.16134_11902E4;
1003 P1 : constant Float_Type'Base := -0.99225_92967E2;
1004 P2 : constant Float_Type'Base := -0.96437_49299E0;
1006 Q0 : constant Float_Type'Base := 0.48402_35707E4;
1007 Q1 : constant Float_Type'Base := 0.22337_72071E4;
1008 Q2 : constant Float_Type'Base := 0.11274_47438E3;
1009 Q3 : constant Float_Type'Base := 0.10000000000E1;
1011 Half_Ln3 : constant Float_Type'Base := 0.54930_61443;
1013 P, Q, R : Float_Type'Base;
1014 Y : Float_Type'Base := abs X;
1015 G : Float_Type'Base := Y * Y;
1017 Float_Type_Digits_15_Or_More : constant Boolean :=
1018 Float_Type'Digits > 14;
1021 if X < Half_Log_Epsilon then
1024 elsif X > -Half_Log_Epsilon then
1027 elsif Y < Sqrt_Epsilon then
1031 and then Float_Type_Digits_15_Or_More
1033 P := (P2 * G + P1) * G + P0;
1034 Q := ((Q3 * G + Q2) * G + Q1) * G + Q0;
1039 return Float_Type'Base (Aux.Tanh (Double (X)));
1043 end Ada.Numerics.Generic_Elementary_Functions;