--- /dev/null
+
+
+------------------------------------------------------------------------
+--
+-- Copyright 1996 by IEEE. All rights reserved.
+--
+-- This source file is an essential part of IEEE Std 1076.2-1996, IEEE Standard
+-- VHDL Mathematical Packages. This source file may not be copied, sold, or
+-- included with software that is sold without written permission from the IEEE
+-- Standards Department. This source file may be used to implement this standard
+-- and may be distributed in compiled form in any manner so long as the
+-- compiled form does not allow direct decompilation of the original source file.
+-- This source file may be copied for individual use between licensed users.
+-- This source file is provided on an AS IS basis. The IEEE disclaims ANY
+-- WARRANTY EXPRESS OR IMPLIED INCLUDING ANY WARRANTY OF MERCHANTABILITY
+-- AND FITNESS FOR USE FOR A PARTICULAR PURPOSE. The user of the source
+-- file shall indemnify and hold IEEE harmless from any damages or liability
+-- arising out of the use thereof.
+--
+-- Title: Standard VHDL Mathematical Packages (IEEE Std 1076.2-1996,
+-- MATH_REAL)
+--
+-- Library: This package shall be compiled into a library
+-- symbolically named IEEE.
+--
+-- Developers: IEEE DASC VHDL Mathematical Packages Working Group
+--
+-- Purpose: This package defines a standard for designers to use in
+-- describing VHDL models that make use of common REAL constants
+-- and common REAL elementary mathematical functions.
+--
+-- Limitation: The values generated by the functions in this package may
+-- vary from platform to platform, and the precision of results
+-- is only guaranteed to be the minimum required by IEEE Std 1076-
+-- 1993.
+--
+-- Notes:
+-- No declarations or definitions shall be included in, or
+-- excluded from, this package.
+-- The "package declaration" defines the types, subtypes, and
+-- declarations of MATH_REAL.
+-- The standard mathematical definition and conventional meaning
+-- of the mathematical functions that are part of this standard
+-- represent the formal semantics of the implementation of the
+-- MATH_REAL package declaration. The purpose of the MATH_REAL
+-- package body is to provide a guideline for implementations to
+-- verify their implementation of MATH_REAL. Tool developers may
+-- choose to implement the package body in the most efficient
+-- manner available to them.
+--
+-- -----------------------------------------------------------------------------
+-- Version : 1.5
+-- Date : 24 July 1996
+-- -----------------------------------------------------------------------------
+
+package MATH_REAL is
+ constant CopyRightNotice: STRING
+ := "Copyright 1996 IEEE. All rights reserved.";
+
+ --
+ -- Constant Definitions
+ --
+ constant MATH_E : REAL := 2.71828_18284_59045_23536;
+ -- Value of e
+ constant MATH_1_OVER_E : REAL := 0.36787_94411_71442_32160;
+ -- Value of 1/e
+ constant MATH_PI : REAL := 3.14159_26535_89793_23846;
+ -- Value of pi
+ constant MATH_2_PI : REAL := 6.28318_53071_79586_47693;
+ -- Value of 2*pi
+ constant MATH_1_OVER_PI : REAL := 0.31830_98861_83790_67154;
+ -- Value of 1/pi
+ constant MATH_PI_OVER_2 : REAL := 1.57079_63267_94896_61923;
+ -- Value of pi/2
+ constant MATH_PI_OVER_3 : REAL := 1.04719_75511_96597_74615;
+ -- Value of pi/3
+ constant MATH_PI_OVER_4 : REAL := 0.78539_81633_97448_30962;
+ -- Value of pi/4
+ constant MATH_3_PI_OVER_2 : REAL := 4.71238_89803_84689_85769;
+ -- Value 3*pi/2
+ constant MATH_LOG_OF_2 : REAL := 0.69314_71805_59945_30942;
+ -- Natural log of 2
+ constant MATH_LOG_OF_10 : REAL := 2.30258_50929_94045_68402;
+ -- Natural log of 10
+ constant MATH_LOG2_OF_E : REAL := 1.44269_50408_88963_4074;
+ -- Log base 2 of e
+ constant MATH_LOG10_OF_E: REAL := 0.43429_44819_03251_82765;
+ -- Log base 10 of e
+ constant MATH_SQRT_2: REAL := 1.41421_35623_73095_04880;
+ -- square root of 2
+ constant MATH_1_OVER_SQRT_2: REAL := 0.70710_67811_86547_52440;
+ -- square root of 1/2
+ constant MATH_SQRT_PI: REAL := 1.77245_38509_05516_02730;
+ -- square root of pi
+ constant MATH_DEG_TO_RAD: REAL := 0.01745_32925_19943_29577;
+ -- Conversion factor from degree to radian
+ constant MATH_RAD_TO_DEG: REAL := 57.29577_95130_82320_87680;
+ -- Conversion factor from radian to degree
+
+ --
+ -- Function Declarations
+ --
+ function SIGN (X: in REAL ) return REAL;
+ -- Purpose:
+ -- Returns 1.0 if X > 0.0; 0.0 if X = 0.0; -1.0 if X < 0.0
+ -- Special values:
+ -- None
+ -- Domain:
+ -- X in REAL
+ -- Error conditions:
+ -- None
+ -- Range:
+ -- ABS(SIGN(X)) <= 1.0
+ -- Notes:
+ -- None
+
+ function CEIL (X : in REAL ) return REAL;
+ -- Purpose:
+ -- Returns smallest INTEGER value (as REAL) not less than X
+ -- Special values:
+ -- None
+ -- Domain:
+ -- X in REAL
+ -- Error conditions:
+ -- None
+ -- Range:
+ -- CEIL(X) is mathematically unbounded
+ -- Notes:
+ -- a) Implementations have to support at least the domain
+ -- ABS(X) < REAL(INTEGER'HIGH)
+
+ function FLOOR (X : in REAL ) return REAL;
+ -- Purpose:
+ -- Returns largest INTEGER value (as REAL) not greater than X
+ -- Special values:
+ -- FLOOR(0.0) = 0.0
+ -- Domain:
+ -- X in REAL
+ -- Error conditions:
+ -- None
+ -- Range:
+ -- FLOOR(X) is mathematically unbounded
+ -- Notes:
+ -- a) Implementations have to support at least the domain
+ -- ABS(X) < REAL(INTEGER'HIGH)
+
+ function ROUND (X : in REAL ) return REAL;
+ -- Purpose:
+ -- Rounds X to the nearest integer value (as real). If X is
+ -- halfway between two integers, rounding is away from 0.0
+ -- Special values:
+ -- ROUND(0.0) = 0.0
+ -- Domain:
+ -- X in REAL
+ -- Error conditions:
+ -- None
+ -- Range:
+ -- ROUND(X) is mathematically unbounded
+ -- Notes:
+ -- a) Implementations have to support at least the domain
+ -- ABS(X) < REAL(INTEGER'HIGH)
+
+ function TRUNC (X : in REAL ) return REAL;
+ -- Purpose:
+ -- Truncates X towards 0.0 and returns truncated value
+ -- Special values:
+ -- TRUNC(0.0) = 0.0
+ -- Domain:
+ -- X in REAL
+ -- Error conditions:
+ -- None
+ -- Range:
+ -- TRUNC(X) is mathematically unbounded
+ -- Notes:
+ -- a) Implementations have to support at least the domain
+ -- ABS(X) < REAL(INTEGER'HIGH)
+
+ function "MOD" (X, Y: in REAL ) return REAL;
+ -- Purpose:
+ -- Returns floating point modulus of X/Y, with the same sign as
+ -- Y, and absolute value less than the absolute value of Y, and
+ -- for some INTEGER value N the result satisfies the relation
+ -- X = Y*N + MOD(X,Y)
+ -- Special values:
+ -- None
+ -- Domain:
+ -- X in REAL; Y in REAL and Y /= 0.0
+ -- Error conditions:
+ -- Error if Y = 0.0
+ -- Range:
+ -- ABS(MOD(X,Y)) < ABS(Y)
+ -- Notes:
+ -- None
+
+ function REALMAX (X, Y : in REAL ) return REAL;
+ -- Purpose:
+ -- Returns the algebraically larger of X and Y
+ -- Special values:
+ -- REALMAX(X,Y) = X when X = Y
+ -- Domain:
+ -- X in REAL; Y in REAL
+ -- Error conditions:
+ -- None
+ -- Range:
+ -- REALMAX(X,Y) is mathematically unbounded
+ -- Notes:
+ -- None
+
+ function REALMIN (X, Y : in REAL ) return REAL;
+ -- Purpose:
+ -- Returns the algebraically smaller of X and Y
+ -- Special values:
+ -- REALMIN(X,Y) = X when X = Y
+ -- Domain:
+ -- X in REAL; Y in REAL
+ -- Error conditions:
+ -- None
+ -- Range:
+ -- REALMIN(X,Y) is mathematically unbounded
+ -- Notes:
+ -- None
+
+ procedure UNIFORM(variable SEED1,SEED2:inout POSITIVE; variable X:out REAL);
+ -- Purpose:
+ -- Returns, in X, a pseudo-random number with uniform
+ -- distribution in the open interval (0.0, 1.0).
+ -- Special values:
+ -- None
+ -- Domain:
+ -- 1 <= SEED1 <= 2147483562; 1 <= SEED2 <= 2147483398
+ -- Error conditions:
+ -- Error if SEED1 or SEED2 outside of valid domain
+ -- Range:
+ -- 0.0 < X < 1.0
+ -- Notes:
+ -- a) The semantics for this function are described by the
+ -- algorithm published by Pierre L'Ecuyer in "Communications
+ -- of the ACM," vol. 31, no. 6, June 1988, pp. 742-774.
+ -- The algorithm is based on the combination of two
+ -- multiplicative linear congruential generators for 32-bit
+ -- platforms.
+ --
+ -- b) Before the first call to UNIFORM, the seed values
+ -- (SEED1, SEED2) have to be initialized to values in the range
+ -- [1, 2147483562] and [1, 2147483398] respectively. The
+ -- seed values are modified after each call to UNIFORM.
+ --
+ -- c) This random number generator is portable for 32-bit
+ -- computers, and it has a period of ~2.30584*(10**18) for each
+ -- set of seed values.
+ --
+ -- d) For information on spectral tests for the algorithm, refer
+ -- to the L'Ecuyer article.
+
+ function SQRT (X : in REAL ) return REAL;
+ -- Purpose:
+ -- Returns square root of X
+ -- Special values:
+ -- SQRT(0.0) = 0.0
+ -- SQRT(1.0) = 1.0
+ -- Domain:
+ -- X >= 0.0
+ -- Error conditions:
+ -- Error if X < 0.0
+ -- Range:
+ -- SQRT(X) >= 0.0
+ -- Notes:
+ -- a) The upper bound of the reachable range of SQRT is
+ -- approximately given by:
+ -- SQRT(X) <= SQRT(REAL'HIGH)
+
+ function CBRT (X : in REAL ) return REAL;
+ -- Purpose:
+ -- Returns cube root of X
+ -- Special values:
+ -- CBRT(0.0) = 0.0
+ -- CBRT(1.0) = 1.0
+ -- CBRT(-1.0) = -1.0
+ -- Domain:
+ -- X in REAL
+ -- Error conditions:
+ -- None
+ -- Range:
+ -- CBRT(X) is mathematically unbounded
+ -- Notes:
+ -- a) The reachable range of CBRT is approximately given by:
+ -- ABS(CBRT(X)) <= CBRT(REAL'HIGH)
+
+ function "**" (X : in INTEGER; Y : in REAL) return REAL;
+ -- Purpose:
+ -- Returns Y power of X ==> X**Y
+ -- Special values:
+ -- X**0.0 = 1.0; X /= 0
+ -- 0**Y = 0.0; Y > 0.0
+ -- X**1.0 = REAL(X); X >= 0
+ -- 1**Y = 1.0
+ -- Domain:
+ -- X > 0
+ -- X = 0 for Y > 0.0
+ -- X < 0 for Y = 0.0
+ -- Error conditions:
+ -- Error if X < 0 and Y /= 0.0
+ -- Error if X = 0 and Y <= 0.0
+ -- Range:
+ -- X**Y >= 0.0
+ -- Notes:
+ -- a) The upper bound of the reachable range for "**" is
+ -- approximately given by:
+ -- X**Y <= REAL'HIGH
+
+ function "**" (X : in REAL; Y : in REAL) return REAL;
+ -- Purpose:
+ -- Returns Y power of X ==> X**Y
+ -- Special values:
+ -- X**0.0 = 1.0; X /= 0.0
+ -- 0.0**Y = 0.0; Y > 0.0
+ -- X**1.0 = X; X >= 0.0
+ -- 1.0**Y = 1.0
+ -- Domain:
+ -- X > 0.0
+ -- X = 0.0 for Y > 0.0
+ -- X < 0.0 for Y = 0.0
+ -- Error conditions:
+ -- Error if X < 0.0 and Y /= 0.0
+ -- Error if X = 0.0 and Y <= 0.0
+ -- Range:
+ -- X**Y >= 0.0
+ -- Notes:
+ -- a) The upper bound of the reachable range for "**" is
+ -- approximately given by:
+ -- X**Y <= REAL'HIGH
+
+ function EXP (X : in REAL ) return REAL;
+ -- Purpose:
+ -- Returns e**X; where e = MATH_E
+ -- Special values:
+ -- EXP(0.0) = 1.0
+ -- EXP(1.0) = MATH_E
+ -- EXP(-1.0) = MATH_1_OVER_E
+ -- EXP(X) = 0.0 for X <= -LOG(REAL'HIGH)
+ -- Domain:
+ -- X in REAL such that EXP(X) <= REAL'HIGH
+ -- Error conditions:
+ -- Error if X > LOG(REAL'HIGH)
+ -- Range:
+ -- EXP(X) >= 0.0
+ -- Notes:
+ -- a) The usable domain of EXP is approximately given by:
+ -- X <= LOG(REAL'HIGH)
+
+ function LOG (X : in REAL ) return REAL;
+ -- Purpose:
+ -- Returns natural logarithm of X
+ -- Special values:
+ -- LOG(1.0) = 0.0
+ -- LOG(MATH_E) = 1.0
+ -- Domain:
+ -- X > 0.0
+ -- Error conditions:
+ -- Error if X <= 0.0
+ -- Range:
+ -- LOG(X) is mathematically unbounded
+ -- Notes:
+ -- a) The reachable range of LOG is approximately given by:
+ -- LOG(0+) <= LOG(X) <= LOG(REAL'HIGH)
+
+ function LOG2 (X : in REAL ) return REAL;
+ -- Purpose:
+ -- Returns logarithm base 2 of X
+ -- Special values:
+ -- LOG2(1.0) = 0.0
+ -- LOG2(2.0) = 1.0
+ -- Domain:
+ -- X > 0.0
+ -- Error conditions:
+ -- Error if X <= 0.0
+ -- Range:
+ -- LOG2(X) is mathematically unbounded
+ -- Notes:
+ -- a) The reachable range of LOG2 is approximately given by:
+ -- LOG2(0+) <= LOG2(X) <= LOG2(REAL'HIGH)
+
+ function LOG10 (X : in REAL ) return REAL;
+ -- Purpose:
+ -- Returns logarithm base 10 of X
+ -- Special values:
+ -- LOG10(1.0) = 0.0
+ -- LOG10(10.0) = 1.0
+ -- Domain:
+ -- X > 0.0
+ -- Error conditions:
+ -- Error if X <= 0.0
+ -- Range:
+ -- LOG10(X) is mathematically unbounded
+ -- Notes:
+ -- a) The reachable range of LOG10 is approximately given by:
+ -- LOG10(0+) <= LOG10(X) <= LOG10(REAL'HIGH)
+
+ function LOG (X: in REAL; BASE: in REAL) return REAL;
+ -- Purpose:
+ -- Returns logarithm base BASE of X
+ -- Special values:
+ -- LOG(1.0, BASE) = 0.0
+ -- LOG(BASE, BASE) = 1.0
+ -- Domain:
+ -- X > 0.0
+ -- BASE > 0.0
+ -- BASE /= 1.0
+ -- Error conditions:
+ -- Error if X <= 0.0
+ -- Error if BASE <= 0.0
+ -- Error if BASE = 1.0
+ -- Range:
+ -- LOG(X, BASE) is mathematically unbounded
+ -- Notes:
+ -- a) When BASE > 1.0, the reachable range of LOG is
+ -- approximately given by:
+ -- LOG(0+, BASE) <= LOG(X, BASE) <= LOG(REAL'HIGH, BASE)
+ -- b) When 0.0 < BASE < 1.0, the reachable range of LOG is
+ -- approximately given by:
+ -- LOG(REAL'HIGH, BASE) <= LOG(X, BASE) <= LOG(0+, BASE)
+
+ function SIN (X : in REAL ) return REAL;
+ -- Purpose:
+ -- Returns sine of X; X in radians
+ -- Special values:
+ -- SIN(X) = 0.0 for X = k*MATH_PI, where k is an INTEGER
+ -- SIN(X) = 1.0 for X = (4*k+1)*MATH_PI_OVER_2, where k is an
+ -- INTEGER
+ -- SIN(X) = -1.0 for X = (4*k+3)*MATH_PI_OVER_2, where k is an
+ -- INTEGER
+ -- Domain:
+ -- X in REAL
+ -- Error conditions:
+ -- None
+ -- Range:
+ -- ABS(SIN(X)) <= 1.0
+ -- Notes:
+ -- a) For larger values of ABS(X), degraded accuracy is allowed.
+
+ function COS ( X : in REAL ) return REAL;
+ -- Purpose:
+ -- Returns cosine of X; X in radians
+ -- Special values:
+ -- COS(X) = 0.0 for X = (2*k+1)*MATH_PI_OVER_2, where k is an
+ -- INTEGER
+ -- COS(X) = 1.0 for X = (2*k)*MATH_PI, where k is an INTEGER
+ -- COS(X) = -1.0 for X = (2*k+1)*MATH_PI, where k is an INTEGER
+ -- Domain:
+ -- X in REAL
+ -- Error conditions:
+ -- None
+ -- Range:
+ -- ABS(COS(X)) <= 1.0
+ -- Notes:
+ -- a) For larger values of ABS(X), degraded accuracy is allowed.
+
+ function TAN (X : in REAL ) return REAL;
+ -- Purpose:
+ -- Returns tangent of X; X in radians
+ -- Special values:
+ -- TAN(X) = 0.0 for X = k*MATH_PI, where k is an INTEGER
+ -- Domain:
+ -- X in REAL and
+ -- X /= (2*k+1)*MATH_PI_OVER_2, where k is an INTEGER
+ -- Error conditions:
+ -- Error if X = ((2*k+1) * MATH_PI_OVER_2), where k is an
+ -- INTEGER
+ -- Range:
+ -- TAN(X) is mathematically unbounded
+ -- Notes:
+ -- a) For larger values of ABS(X), degraded accuracy is allowed.
+
+ function ARCSIN (X : in REAL ) return REAL;
+ -- Purpose:
+ -- Returns inverse sine of X
+ -- Special values:
+ -- ARCSIN(0.0) = 0.0
+ -- ARCSIN(1.0) = MATH_PI_OVER_2
+ -- ARCSIN(-1.0) = -MATH_PI_OVER_2
+ -- Domain:
+ -- ABS(X) <= 1.0
+ -- Error conditions:
+ -- Error if ABS(X) > 1.0
+ -- Range:
+ -- ABS(ARCSIN(X) <= MATH_PI_OVER_2
+ -- Notes:
+ -- None
+
+ function ARCCOS (X : in REAL ) return REAL;
+ -- Purpose:
+ -- Returns inverse cosine of X
+ -- Special values:
+ -- ARCCOS(1.0) = 0.0
+ -- ARCCOS(0.0) = MATH_PI_OVER_2
+ -- ARCCOS(-1.0) = MATH_PI
+ -- Domain:
+ -- ABS(X) <= 1.0
+ -- Error conditions:
+ -- Error if ABS(X) > 1.0
+ -- Range:
+ -- 0.0 <= ARCCOS(X) <= MATH_PI
+ -- Notes:
+ -- None
+
+ function ARCTAN (Y : in REAL) return REAL;
+ -- Purpose:
+ -- Returns the value of the angle in radians of the point
+ -- (1.0, Y), which is in rectangular coordinates
+ -- Special values:
+ -- ARCTAN(0.0) = 0.0
+ -- Domain:
+ -- Y in REAL
+ -- Error conditions:
+ -- None
+ -- Range:
+ -- ABS(ARCTAN(Y)) <= MATH_PI_OVER_2
+ -- Notes:
+ -- None
+
+ function ARCTAN (Y : in REAL; X : in REAL) return REAL;
+ -- Purpose:
+ -- Returns the principal value of the angle in radians of
+ -- the point (X, Y), which is in rectangular coordinates
+ -- Special values:
+ -- ARCTAN(0.0, X) = 0.0 if X > 0.0
+ -- ARCTAN(0.0, X) = MATH_PI if X < 0.0
+ -- ARCTAN(Y, 0.0) = MATH_PI_OVER_2 if Y > 0.0
+ -- ARCTAN(Y, 0.0) = -MATH_PI_OVER_2 if Y < 0.0
+ -- Domain:
+ -- Y in REAL
+ -- X in REAL, X /= 0.0 when Y = 0.0
+ -- Error conditions:
+ -- Error if X = 0.0 and Y = 0.0
+ -- Range:
+ -- -MATH_PI < ARCTAN(Y,X) <= MATH_PI
+ -- Notes:
+ -- None
+
+ function SINH (X : in REAL) return REAL;
+ -- Purpose:
+ -- Returns hyperbolic sine of X
+ -- Special values:
+ -- SINH(0.0) = 0.0
+ -- Domain:
+ -- X in REAL
+ -- Error conditions:
+ -- None
+ -- Range:
+ -- SINH(X) is mathematically unbounded
+ -- Notes:
+ -- a) The usable domain of SINH is approximately given by:
+ -- ABS(X) <= LOG(REAL'HIGH)
+
+
+ function COSH (X : in REAL) return REAL;
+ -- Purpose:
+ -- Returns hyperbolic cosine of X
+ -- Special values:
+ -- COSH(0.0) = 1.0
+ -- Domain:
+ -- X in REAL
+ -- Error conditions:
+ -- None
+ -- Range:
+ -- COSH(X) >= 1.0
+ -- Notes:
+ -- a) The usable domain of COSH is approximately given by:
+ -- ABS(X) <= LOG(REAL'HIGH)
+
+ function TANH (X : in REAL) return REAL;
+ -- Purpose:
+ -- Returns hyperbolic tangent of X
+ -- Special values:
+ -- TANH(0.0) = 0.0
+ -- Domain:
+ -- X in REAL
+ -- Error conditions:
+ -- None
+ -- Range:
+ -- ABS(TANH(X)) <= 1.0
+ -- Notes:
+ -- None
+
+ function ARCSINH (X : in REAL) return REAL;
+ -- Purpose:
+ -- Returns inverse hyperbolic sine of X
+ -- Special values:
+ -- ARCSINH(0.0) = 0.0
+ -- Domain:
+ -- X in REAL
+ -- Error conditions:
+ -- None
+ -- Range:
+ -- ARCSINH(X) is mathematically unbounded
+ -- Notes:
+ -- a) The reachable range of ARCSINH is approximately given by:
+ -- ABS(ARCSINH(X)) <= LOG(REAL'HIGH)
+
+ function ARCCOSH (X : in REAL) return REAL;
+ -- Purpose:
+ -- Returns inverse hyperbolic cosine of X
+ -- Special values:
+ -- ARCCOSH(1.0) = 0.0
+ -- Domain:
+ -- X >= 1.0
+ -- Error conditions:
+ -- Error if X < 1.0
+ -- Range:
+ -- ARCCOSH(X) >= 0.0
+ -- Notes:
+ -- a) The upper bound of the reachable range of ARCCOSH is
+ -- approximately given by: ARCCOSH(X) <= LOG(REAL'HIGH)
+
+ function ARCTANH (X : in REAL) return REAL;
+ -- Purpose:
+ -- Returns inverse hyperbolic tangent of X
+ -- Special values:
+ -- ARCTANH(0.0) = 0.0
+ -- Domain:
+ -- ABS(X) < 1.0
+ -- Error conditions:
+ -- Error if ABS(X) >= 1.0
+ -- Range:
+ -- ARCTANH(X) is mathematically unbounded
+ -- Notes:
+ -- a) The reachable range of ARCTANH is approximately given by:
+ -- ABS(ARCTANH(X)) < LOG(REAL'HIGH)
+
+end MATH_REAL;
+
+
+
+------------------------------------------------------------------------
+--
+-- Copyright 1996 by IEEE. All rights reserved.
+
+-- This source file is an informative part of IEEE Std 1076.2-1996, IEEE Standard
+-- VHDL Mathematical Packages. This source file may not be copied, sold, or
+-- included with software that is sold without written permission from the IEEE
+-- Standards Department. This source file may be used to implement this standard
+-- and may be distributed in compiled form in any manner so long as the
+-- compiled form does not allow direct decompilation of the original source file.
+-- This source file may be copied for individual use between licensed users.
+-- This source file is provided on an AS IS basis. The IEEE disclaims ANY
+-- WARRANTY EXPRESS OR IMPLIED INCLUDING ANY WARRANTY OF MERCHANTABILITY
+-- AND FITNESS FOR USE FOR A PARTICULAR PURPOSE. The user of the source
+-- file shall indemnify and hold IEEE harmless from any damages or liability
+-- arising out of the use thereof.
+
+--
+-- Title: Standard VHDL Mathematical Packages (IEEE Std 1076.2-1996,
+-- MATH_REAL)
+--
+-- Library: This package shall be compiled into a library
+-- symbolically named IEEE.
+--
+-- Developers: IEEE DASC VHDL Mathematical Packages Working Group
+--
+-- Purpose: This package body is a nonnormative implementation of the
+-- functionality defined in the MATH_REAL package declaration.
+--
+-- Limitation: The values generated by the functions in this package may
+-- vary from platform to platform, and the precision of results
+-- is only guaranteed to be the minimum required by IEEE Std 1076
+-- -1993.
+--
+-- Notes:
+-- The "package declaration" defines the types, subtypes, and
+-- declarations of MATH_REAL.
+-- The standard mathematical definition and conventional meaning
+-- of the mathematical functions that are part of this standard
+-- represent the formal semantics of the implementation of the
+-- MATH_REAL package declaration. The purpose of the MATH_REAL
+-- package body is to clarify such semantics and provide a
+-- guideline for implementations to verify their implementation
+-- of MATH_REAL. Tool developers may choose to implement
+-- the package body in the most efficient manner available to them.
+--
+-- -----------------------------------------------------------------------------
+-- Version : 1.5
+-- Date : 24 July 1996
+-- -----------------------------------------------------------------------------
+
+package body MATH_REAL is
+
+ --
+ -- Local Constants for Use in the Package Body Only
+ --
+ constant MATH_E_P2 : REAL := 7.38905_60989_30650; -- e**2
+ constant MATH_E_P10 : REAL := 22026.46579_48067_17; -- e**10
+ constant MATH_EIGHT_PI : REAL := 25.13274_12287_18345_90770_115; --8*pi
+ constant MAX_ITER: INTEGER := 27; -- Maximum precision factor for cordic
+ constant MAX_COUNT: INTEGER := 150; -- Maximum count for number of tries
+ constant BASE_EPS: REAL := 0.00001; -- Factor for convergence criteria
+ constant KC : REAL := 6.0725293500888142e-01; -- Constant for cordic
+
+ --
+ -- Local Type Declarations for Cordic Operations
+ --
+ type REAL_VECTOR is array (NATURAL range <>) of REAL;
+ type NATURAL_VECTOR is array (NATURAL range <>) of NATURAL;
+ subtype REAL_VECTOR_N is REAL_VECTOR (0 to MAX_ITER);
+ subtype REAL_ARR_2 is REAL_VECTOR (0 to 1);
+ subtype REAL_ARR_3 is REAL_VECTOR (0 to 2);
+ subtype QUADRANT is INTEGER range 0 to 3;
+ type CORDIC_MODE_TYPE is (ROTATION, VECTORING);
+
+ --
+ -- Auxiliary Functions for Cordic Algorithms
+ --
+ function POWER_OF_2_SERIES (D : in NATURAL_VECTOR; INITIAL_VALUE : in REAL;
+ NUMBER_OF_VALUES : in NATURAL) return REAL_VECTOR is
+ -- Description:
+ -- Returns power of two for a vector of values
+ -- Notes:
+ -- None
+ --
+ variable V : REAL_VECTOR (0 to NUMBER_OF_VALUES);
+ variable TEMP : REAL := INITIAL_VALUE;
+ variable FLAG : BOOLEAN := TRUE;
+ begin
+ for I in 0 to NUMBER_OF_VALUES loop
+ V(I) := TEMP;
+ for P in D'RANGE loop
+ if I = D(P) then
+ FLAG := FALSE;
+ exit;
+ end if;
+ end loop;
+ if FLAG then
+ TEMP := TEMP/2.0;
+ end if;
+ FLAG := TRUE;
+ end loop;
+ return V;
+ end POWER_OF_2_SERIES;
+
+
+ constant TWO_AT_MINUS : REAL_VECTOR := POWER_OF_2_SERIES(
+ NATURAL_VECTOR'(100, 90),1.0,
+ MAX_ITER);
+
+ constant EPSILON : REAL_VECTOR_N := (
+ 7.8539816339744827e-01,
+ 4.6364760900080606e-01,
+ 2.4497866312686413e-01,
+ 1.2435499454676144e-01,
+ 6.2418809995957351e-02,
+ 3.1239833430268277e-02,
+ 1.5623728620476830e-02,
+ 7.8123410601011116e-03,
+ 3.9062301319669717e-03,
+ 1.9531225164788189e-03,
+ 9.7656218955931937e-04,
+ 4.8828121119489829e-04,
+ 2.4414062014936175e-04,
+ 1.2207031189367021e-04,
+ 6.1035156174208768e-05,
+ 3.0517578115526093e-05,
+ 1.5258789061315760e-05,
+ 7.6293945311019699e-06,
+ 3.8146972656064960e-06,
+ 1.9073486328101870e-06,
+ 9.5367431640596080e-07,
+ 4.7683715820308876e-07,
+ 2.3841857910155801e-07,
+ 1.1920928955078067e-07,
+ 5.9604644775390553e-08,
+ 2.9802322387695303e-08,
+ 1.4901161193847654e-08,
+ 7.4505805969238281e-09
+ );
+
+ function CORDIC ( X0 : in REAL;
+ Y0 : in REAL;
+ Z0 : in REAL;
+ N : in NATURAL; -- Precision factor
+ CORDIC_MODE : in CORDIC_MODE_TYPE -- Rotation (Z -> 0)
+ -- or vectoring (Y -> 0)
+ ) return REAL_ARR_3 is
+ -- Description:
+ -- Compute cordic values
+ -- Notes:
+ -- None
+ variable X : REAL := X0;
+ variable Y : REAL := Y0;
+ variable Z : REAL := Z0;
+ variable X_TEMP : REAL;
+ begin
+ if CORDIC_MODE = ROTATION then
+ for K in 0 to N loop
+ X_TEMP := X;
+ if ( Z >= 0.0) then
+ X := X - Y * TWO_AT_MINUS(K);
+ Y := Y + X_TEMP * TWO_AT_MINUS(K);
+ Z := Z - EPSILON(K);
+ else
+ X := X + Y * TWO_AT_MINUS(K);
+ Y := Y - X_TEMP * TWO_AT_MINUS(K);
+ Z := Z + EPSILON(K);
+ end if;
+ end loop;
+ else
+ for K in 0 to N loop
+ X_TEMP := X;
+ if ( Y < 0.0) then
+ X := X - Y * TWO_AT_MINUS(K);
+ Y := Y + X_TEMP * TWO_AT_MINUS(K);
+ Z := Z - EPSILON(K);
+ else
+ X := X + Y * TWO_AT_MINUS(K);
+ Y := Y - X_TEMP * TWO_AT_MINUS(K);
+ Z := Z + EPSILON(K);
+ end if;
+ end loop;
+ end if;
+ return REAL_ARR_3'(X, Y, Z);
+ end CORDIC;
+
+ --
+ -- Bodies for Global Mathematical Functions Start Here
+ --
+ function SIGN (X: in REAL ) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- None
+ begin
+ if ( X > 0.0 ) then
+ return 1.0;
+ elsif ( X < 0.0 ) then
+ return -1.0;
+ else
+ return 0.0;
+ end if;
+ end SIGN;
+
+ function CEIL (X : in REAL ) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) No conversion to an INTEGER type is expected, so truncate
+ -- cannot overflow for large arguments
+ -- b) The domain supported by this function is X <= LARGE
+ -- c) Returns X if ABS(X) >= LARGE
+
+ constant LARGE: REAL := REAL(INTEGER'HIGH);
+ variable RD: REAL;
+
+ begin
+ if ABS(X) >= LARGE then
+ return X;
+ end if;
+
+ RD := REAL ( INTEGER(X));
+ if RD = X then
+ return X;
+ end if;
+
+ if X > 0.0 then
+ if RD >= X then
+ return RD;
+ else
+ return RD + 1.0;
+ end if;
+ elsif X = 0.0 then
+ return 0.0;
+ else
+ if RD <= X then
+ return RD + 1.0;
+ else
+ return RD;
+ end if;
+ end if;
+ end CEIL;
+
+ function FLOOR (X : in REAL ) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) No conversion to an INTEGER type is expected, so truncate
+ -- cannot overflow for large arguments
+ -- b) The domain supported by this function is ABS(X) <= LARGE
+ -- c) Returns X if ABS(X) >= LARGE
+
+ constant LARGE: REAL := REAL(INTEGER'HIGH);
+ variable RD: REAL;
+
+ begin
+ if ABS( X ) >= LARGE then
+ return X;
+ end if;
+
+ RD := REAL ( INTEGER(X));
+ if RD = X then
+ return X;
+ end if;
+
+ if X > 0.0 then
+ if RD <= X then
+ return RD;
+ else
+ return RD - 1.0;
+ end if;
+ elsif X = 0.0 then
+ return 0.0;
+ else
+ if RD >= X then
+ return RD - 1.0;
+ else
+ return RD;
+ end if;
+ end if;
+ end FLOOR;
+
+ function ROUND (X : in REAL ) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) Returns 0.0 if X = 0.0
+ -- b) Returns FLOOR(X + 0.5) if X > 0
+ -- c) Returns CEIL(X - 0.5) if X < 0
+
+ begin
+ if X > 0.0 then
+ return FLOOR(X + 0.5);
+ elsif X < 0.0 then
+ return CEIL( X - 0.5);
+ else
+ return 0.0;
+ end if;
+ end ROUND;
+
+ function TRUNC (X : in REAL ) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) Returns 0.0 if X = 0.0
+ -- b) Returns FLOOR(X) if X > 0
+ -- c) Returns CEIL(X) if X < 0
+
+ begin
+ if X > 0.0 then
+ return FLOOR(X);
+ elsif X < 0.0 then
+ return CEIL( X);
+ else
+ return 0.0;
+ end if;
+ end TRUNC;
+
+
+
+
+ function "MOD" (X, Y: in REAL ) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) Returns 0.0 on error
+
+ variable XNEGATIVE : BOOLEAN := X < 0.0;
+ variable YNEGATIVE : BOOLEAN := Y < 0.0;
+ variable VALUE : REAL;
+ begin
+ -- Check validity of input arguments
+ if (Y = 0.0) then
+ assert FALSE
+ report "MOD(X, 0.0) is undefined"
+ severity ERROR;
+ return 0.0;
+ end if;
+
+ -- Compute value
+ if ( XNEGATIVE ) then
+ if ( YNEGATIVE ) then
+ VALUE := X + (FLOOR(ABS(X)/ABS(Y)))*ABS(Y);
+ else
+ VALUE := X + (CEIL(ABS(X)/ABS(Y)))*ABS(Y);
+ end if;
+ else
+ if ( YNEGATIVE ) then
+ VALUE := X - (CEIL(ABS(X)/ABS(Y)))*ABS(Y);
+ else
+ VALUE := X - (FLOOR(ABS(X)/ABS(Y)))*ABS(Y);
+ end if;
+ end if;
+
+ return VALUE;
+ end "MOD";
+
+
+ function REALMAX (X, Y : in REAL ) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) REALMAX(X,Y) = X when X = Y
+ --
+ begin
+ if X >= Y then
+ return X;
+ else
+ return Y;
+ end if;
+ end REALMAX;
+
+ function REALMIN (X, Y : in REAL ) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) REALMIN(X,Y) = X when X = Y
+ --
+ begin
+ if X <= Y then
+ return X;
+ else
+ return Y;
+ end if;
+ end REALMIN;
+
+
+ procedure UNIFORM(variable SEED1,SEED2:inout POSITIVE;variable X:out REAL)
+ is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) Returns 0.0 on error
+ --
+ variable Z, K: INTEGER;
+ variable TSEED1 : INTEGER := INTEGER'(SEED1);
+ variable TSEED2 : INTEGER := INTEGER'(SEED2);
+ begin
+ -- Check validity of arguments
+ if SEED1 > 2147483562 then
+ assert FALSE
+ report "SEED1 > 2147483562 in UNIFORM"
+ severity ERROR;
+ X := 0.0;
+ return;
+ end if;
+
+ if SEED2 > 2147483398 then
+ assert FALSE
+ report "SEED2 > 2147483398 in UNIFORM"
+ severity ERROR;
+ X := 0.0;
+ return;
+ end if;
+
+ -- Compute new seed values and pseudo-random number
+ K := TSEED1/53668;
+ TSEED1 := 40014 * (TSEED1 - K * 53668) - K * 12211;
+
+ if TSEED1 < 0 then
+ TSEED1 := TSEED1 + 2147483563;
+ end if;
+
+ K := TSEED2/52774;
+ TSEED2 := 40692 * (TSEED2 - K * 52774) - K * 3791;
+
+ if TSEED2 < 0 then
+ TSEED2 := TSEED2 + 2147483399;
+ end if;
+
+ Z := TSEED1 - TSEED2;
+ if Z < 1 then
+ Z := Z + 2147483562;
+ end if;
+
+ -- Get output values
+ SEED1 := POSITIVE'(TSEED1);
+ SEED2 := POSITIVE'(TSEED2);
+ X := REAL(Z)*4.656613e-10;
+ end UNIFORM;
+
+
+
+ function SQRT (X : in REAL ) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) Uses the Newton-Raphson approximation:
+ -- F(n+1) = 0.5*[F(n) + x/F(n)]
+ -- b) Returns 0.0 on error
+ --
+
+ constant EPS : REAL := BASE_EPS*BASE_EPS; -- Convergence factor
+
+ variable INIVAL: REAL;
+ variable OLDVAL : REAL ;
+ variable NEWVAL : REAL ;
+ variable COUNT : INTEGER := 1;
+
+ begin
+ -- Check validity of argument
+ if ( X < 0.0 ) then
+ assert FALSE
+ report "X < 0.0 in SQRT(X)"
+ severity ERROR;
+ return 0.0;
+ end if;
+
+ -- Get the square root for special cases
+ if X = 0.0 then
+ return 0.0;
+ else
+ if ( X = 1.0 ) then
+ return 1.0;
+ end if;
+ end if;
+
+ -- Get the square root for general cases
+ INIVAL := EXP(LOG(X)*(0.5)); -- Mathematically correct but imprecise
+ OLDVAL := INIVAL;
+ NEWVAL := (X/OLDVAL + OLDVAL)*0.5;
+
+ -- Check for relative and absolute error and max count
+ while ( ( (ABS((NEWVAL -OLDVAL)/NEWVAL) > EPS) OR
+ (ABS(NEWVAL - OLDVAL) > EPS) ) AND
+ (COUNT < MAX_COUNT) ) loop
+ OLDVAL := NEWVAL;
+ NEWVAL := (X/OLDVAL + OLDVAL)*0.5;
+ COUNT := COUNT + 1;
+ end loop;
+ return NEWVAL;
+ end SQRT;
+
+ function CBRT (X : in REAL ) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) Uses the Newton-Raphson approximation:
+ -- F(n+1) = (1/3)*[2*F(n) + x/F(n)**2];
+ --
+ constant EPS : REAL := BASE_EPS*BASE_EPS;
+
+ variable INIVAL: REAL;
+ variable XLOCAL : REAL := X;
+ variable NEGATIVE : BOOLEAN := X < 0.0;
+ variable OLDVAL : REAL ;
+ variable NEWVAL : REAL ;
+ variable COUNT : INTEGER := 1;
+
+ begin
+
+ -- Compute root for special cases
+ if X = 0.0 then
+ return 0.0;
+ elsif ( X = 1.0 ) then
+ return 1.0;
+ else
+ if X = -1.0 then
+ return -1.0;
+ end if;
+ end if;
+
+ -- Compute root for general cases
+ if NEGATIVE then
+ XLOCAL := -X;
+ end if;
+
+ INIVAL := EXP(LOG(XLOCAL)/(3.0)); -- Mathematically correct but
+ -- imprecise
+ OLDVAL := INIVAL;
+ NEWVAL := (XLOCAL/(OLDVAL*OLDVAL) + 2.0*OLDVAL)/3.0;
+
+ -- Check for relative and absolute errors and max count
+ while ( ( (ABS((NEWVAL -OLDVAL)/NEWVAL) > EPS ) OR
+ (ABS(NEWVAL - OLDVAL) > EPS ) ) AND
+ ( COUNT < MAX_COUNT ) ) loop
+ OLDVAL := NEWVAL;
+ NEWVAL :=(XLOCAL/(OLDVAL*OLDVAL) + 2.0*OLDVAL)/3.0;
+ COUNT := COUNT + 1;
+ end loop;
+
+ if NEGATIVE then
+ NEWVAL := -NEWVAL;
+ end if;
+
+ return NEWVAL;
+ end CBRT;
+
+ function "**" (X : in INTEGER; Y : in REAL) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) Returns 0.0 on error condition
+
+ begin
+ -- Check validity of argument
+ if ( ( X < 0 ) and ( Y /= 0.0 ) ) then
+ assert FALSE
+ report "X < 0 and Y /= 0.0 in X**Y"
+ severity ERROR;
+ return 0.0;
+ end if;
+
+ if ( ( X = 0 ) and ( Y <= 0.0 ) ) then
+ assert FALSE
+ report "X = 0 and Y <= 0.0 in X**Y"
+ severity ERROR;
+ return 0.0;
+ end if;
+
+ -- Get value for special cases
+ if ( X = 0 and Y > 0.0 ) then
+ return 0.0;
+ end if;
+
+ if ( X = 1 ) then
+ return 1.0;
+ end if;
+
+ if ( Y = 0.0 and X /= 0 ) then
+ return 1.0;
+ end if;
+
+ if ( Y = 1.0) then
+ return (REAL(X));
+ end if;
+
+ -- Get value for general case
+ return EXP (Y * LOG (REAL(X)));
+ end "**";
+
+ function "**" (X : in REAL; Y : in REAL) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) Returns 0.0 on error condition
+
+ begin
+ -- Check validity of argument
+ if ( ( X < 0.0 ) and ( Y /= 0.0 ) ) then
+ assert FALSE
+ report "X < 0.0 and Y /= 0.0 in X**Y"
+ severity ERROR;
+ return 0.0;
+ end if;
+
+ if ( ( X = 0.0 ) and ( Y <= 0.0 ) ) then
+ assert FALSE
+ report "X = 0.0 and Y <= 0.0 in X**Y"
+ severity ERROR;
+ return 0.0;
+ end if;
+
+ -- Get value for special cases
+ if ( X = 0.0 and Y > 0.0 ) then
+ return 0.0;
+ end if;
+
+ if ( X = 1.0 ) then
+ return 1.0;
+ end if;
+
+ if ( Y = 0.0 and X /= 0.0 ) then
+ return 1.0;
+ end if;
+
+ if ( Y = 1.0) then
+ return (X);
+ end if;
+
+ -- Get value for general case
+ return EXP (Y * LOG (X));
+ end "**";
+
+ function EXP (X : in REAL ) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) This function computes the exponential using the following
+ -- series:
+ -- exp(x) = 1 + x + x**2/2! + x**3/3! + ... ; |x| < 1.0
+ -- and reduces argument X to take advantage of exp(x+y) =
+ -- exp(x)*exp(y)
+ --
+ -- b) This implementation limits X to be less than LOG(REAL'HIGH)
+ -- to avoid overflow. Returns REAL'HIGH when X reaches that
+ -- limit
+ --
+ constant EPS : REAL := BASE_EPS*BASE_EPS*BASE_EPS;-- Precision criteria
+
+ variable RECIPROCAL: BOOLEAN := X < 0.0;-- Check sign of argument
+ variable XLOCAL : REAL := ABS(X); -- Use positive value
+ variable OLDVAL: REAL ;
+ variable COUNT: INTEGER ;
+ variable NEWVAL: REAL ;
+ variable LAST_TERM: REAL ;
+ variable FACTOR : REAL := 1.0;
+
+ begin
+ -- Compute value for special cases
+ if X = 0.0 then
+ return 1.0;
+ end if;
+
+ if XLOCAL = 1.0 then
+ if RECIPROCAL then
+ return MATH_1_OVER_E;
+ else
+ return MATH_E;
+ end if;
+ end if;
+
+ if XLOCAL = 2.0 then
+ if RECIPROCAL then
+ return 1.0/MATH_E_P2;
+ else
+ return MATH_E_P2;
+ end if;
+ end if;
+
+ if XLOCAL = 10.0 then
+ if RECIPROCAL then
+ return 1.0/MATH_E_P10;
+ else
+ return MATH_E_P10;
+ end if;
+ end if;
+
+ if XLOCAL > LOG(REAL'HIGH) then
+ if RECIPROCAL then
+ return 0.0;
+ else
+ assert FALSE
+ report "X > LOG(REAL'HIGH) in EXP(X)"
+ severity NOTE;
+ return REAL'HIGH;
+ end if;
+ end if;
+
+ -- Reduce argument to ABS(X) < 1.0
+ while XLOCAL > 10.0 loop
+ XLOCAL := XLOCAL - 10.0;
+ FACTOR := FACTOR*MATH_E_P10;
+ end loop;
+
+ while XLOCAL > 1.0 loop
+ XLOCAL := XLOCAL - 1.0;
+ FACTOR := FACTOR*MATH_E;
+ end loop;
+
+ -- Compute value for case 0 < XLOCAL < 1
+ OLDVAL := 1.0;
+ LAST_TERM := XLOCAL;
+ NEWVAL:= OLDVAL + LAST_TERM;
+ COUNT := 2;
+
+ -- Check for relative and absolute errors and max count
+ while ( ( (ABS((NEWVAL - OLDVAL)/NEWVAL) > EPS) OR
+ (ABS(NEWVAL - OLDVAL) > EPS) ) AND
+ (COUNT < MAX_COUNT ) ) loop
+ OLDVAL := NEWVAL;
+ LAST_TERM := LAST_TERM*(XLOCAL / (REAL(COUNT)));
+ NEWVAL := OLDVAL + LAST_TERM;
+ COUNT := COUNT + 1;
+ end loop;
+
+ -- Compute final value using exp(x+y) = exp(x)*exp(y)
+ NEWVAL := NEWVAL*FACTOR;
+
+ if RECIPROCAL then
+ NEWVAL := 1.0/NEWVAL;
+ end if;
+
+ return NEWVAL;
+ end EXP;
+
+
+ --
+ -- Auxiliary Functions to Compute LOG
+ --
+ function ILOGB(X: in REAL) return INTEGER IS
+ -- Description:
+ -- Returns n such that -1 <= ABS(X)/2^n < 2
+ -- Notes:
+ -- None
+
+ variable N: INTEGER := 0;
+ variable Y: REAL := ABS(X);
+
+ begin
+ if(Y = 1.0 or Y = 0.0) then
+ return 0;
+ end if;
+
+ if( Y > 1.0) then
+ while Y >= 2.0 loop
+ Y := Y/2.0;
+ N := N+1;
+ end loop;
+ return N;
+ end if;
+
+ -- O < Y < 1
+ while Y < 1.0 loop
+ Y := Y*2.0;
+ N := N -1;
+ end loop;
+ return N;
+ end ILOGB;
+
+ function LDEXP(X: in REAL; N: in INTEGER) RETURN REAL IS
+ -- Description:
+ -- Returns X*2^n
+ -- Notes:
+ -- None
+ begin
+ return X*(2.0 ** N);
+ end LDEXP;
+
+ function LOG (X : in REAL ) return REAL IS
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ --
+ -- Notes:
+ -- a) Returns REAL'LOW on error
+ --
+ -- Copyright (c) 1992 Regents of the University of California.
+ -- All rights reserved.
+ --
+ -- Redistribution and use in source and binary forms, with or without
+ -- modification, are permitted provided that the following conditions
+ -- are met:
+ -- 1. Redistributions of source code must retain the above copyright
+ -- notice, this list of conditions and the following disclaimer.
+ -- 2. Redistributions in binary form must reproduce the above copyright
+ -- notice, this list of conditions and the following disclaimer in the
+ -- documentation and/or other materials provided with the distribution.
+ -- 3. All advertising materials mentioning features or use of this
+ -- software must display the following acknowledgement:
+ -- This product includes software developed by the University of
+ -- California, Berkeley and its contributors.
+ -- 4. Neither the name of the University nor the names of its
+ -- contributors may be used to endorse or promote products derived
+ -- from this software without specific prior written permission.
+ --
+ -- THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS''
+ -- AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO,
+ -- THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
+ -- PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR
+ -- CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
+ -- EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
+ -- PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
+ -- PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY
+ -- OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
+ -- (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE
+ -- USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH
+ -- DAMAGE.
+ --
+ -- NOTE: This VHDL version was generated using the C version of the
+ -- original function by the IEEE VHDL Mathematical Package
+ -- Working Group (CS/JT)
+
+ constant N: INTEGER := 128;
+
+ -- Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
+ -- Used for generation of extend precision logarithms.
+ -- The constant 35184372088832 is 2^45, so the divide is exact.
+ -- It ensures correct reading of logF_head, even for inaccurate
+ -- decimal-to-binary conversion routines. (Everybody gets the
+ -- right answer for INTEGERs less than 2^53.)
+ -- Values for LOG(F) were generated using error < 10^-57 absolute
+ -- with the bc -l package.
+
+ type REAL_VECTOR is array (NATURAL range <>) of REAL;
+
+ constant A1:REAL := 0.08333333333333178827;
+ constant A2:REAL := 0.01250000000377174923;
+ constant A3:REAL := 0.002232139987919447809;
+ constant A4:REAL := 0.0004348877777076145742;
+
+ constant LOGF_HEAD: REAL_VECTOR(0 TO N) := (
+ 0.0,
+ 0.007782140442060381246,
+ 0.015504186535963526694,
+ 0.023167059281547608406,
+ 0.030771658666765233647,
+ 0.038318864302141264488,
+ 0.045809536031242714670,
+ 0.053244514518837604555,
+ 0.060624621816486978786,
+ 0.067950661908525944454,
+ 0.075223421237524235039,
+ 0.082443669210988446138,
+ 0.089612158689760690322,
+ 0.096729626458454731618,
+ 0.103796793681567578460,
+ 0.110814366340264314203,
+ 0.117783035656430001836,
+ 0.124703478501032805070,
+ 0.131576357788617315236,
+ 0.138402322859292326029,
+ 0.145182009844575077295,
+ 0.151916042025732167530,
+ 0.158605030176659056451,
+ 0.165249572895390883786,
+ 0.171850256926518341060,
+ 0.178407657472689606947,
+ 0.184922338493834104156,
+ 0.191394852999565046047,
+ 0.197825743329758552135,
+ 0.204215541428766300668,
+ 0.210564769107350002741,
+ 0.216873938300523150246,
+ 0.223143551314024080056,
+ 0.229374101064877322642,
+ 0.235566071312860003672,
+ 0.241719936886966024758,
+ 0.247836163904594286577,
+ 0.253915209980732470285,
+ 0.259957524436686071567,
+ 0.265963548496984003577,
+ 0.271933715484010463114,
+ 0.277868451003087102435,
+ 0.283768173130738432519,
+ 0.289633292582948342896,
+ 0.295464212893421063199,
+ 0.301261330578199704177,
+ 0.307025035294827830512,
+ 0.312755710004239517729,
+ 0.318453731118097493890,
+ 0.324119468654316733591,
+ 0.329753286372579168528,
+ 0.335355541920762334484,
+ 0.340926586970454081892,
+ 0.346466767346100823488,
+ 0.351976423156884266063,
+ 0.357455888922231679316,
+ 0.362905493689140712376,
+ 0.368325561158599157352,
+ 0.373716409793814818840,
+ 0.379078352934811846353,
+ 0.384411698910298582632,
+ 0.389716751140440464951,
+ 0.394993808240542421117,
+ 0.400243164127459749579,
+ 0.405465108107819105498,
+ 0.410659924985338875558,
+ 0.415827895143593195825,
+ 0.420969294644237379543,
+ 0.426084395310681429691,
+ 0.431173464818130014464,
+ 0.436236766774527495726,
+ 0.441274560805140936281,
+ 0.446287102628048160113,
+ 0.451274644139630254358,
+ 0.456237433481874177232,
+ 0.461175715122408291790,
+ 0.466089729924533457960,
+ 0.470979715219073113985,
+ 0.475845904869856894947,
+ 0.480688529345570714212,
+ 0.485507815781602403149,
+ 0.490303988045525329653,
+ 0.495077266798034543171,
+ 0.499827869556611403822,
+ 0.504556010751912253908,
+ 0.509261901790523552335,
+ 0.513945751101346104405,
+ 0.518607764208354637958,
+ 0.523248143765158602036,
+ 0.527867089620485785417,
+ 0.532464798869114019908,
+ 0.537041465897345915436,
+ 0.541597282432121573947,
+ 0.546132437597407260909,
+ 0.550647117952394182793,
+ 0.555141507540611200965,
+ 0.559615787935399566777,
+ 0.564070138285387656651,
+ 0.568504735352689749561,
+ 0.572919753562018740922,
+ 0.577315365035246941260,
+ 0.581691739635061821900,
+ 0.586049045003164792433,
+ 0.590387446602107957005,
+ 0.594707107746216934174,
+ 0.599008189645246602594,
+ 0.603290851438941899687,
+ 0.607555250224322662688,
+ 0.611801541106615331955,
+ 0.616029877215623855590,
+ 0.620240409751204424537,
+ 0.624433288012369303032,
+ 0.628608659422752680256,
+ 0.632766669570628437213,
+ 0.636907462236194987781,
+ 0.641031179420679109171,
+ 0.645137961373620782978,
+ 0.649227946625615004450,
+ 0.653301272011958644725,
+ 0.657358072709030238911,
+ 0.661398482245203922502,
+ 0.665422632544505177065,
+ 0.669430653942981734871,
+ 0.673422675212350441142,
+ 0.677398823590920073911,
+ 0.681359224807238206267,
+ 0.685304003098281100392,
+ 0.689233281238557538017,
+ 0.693147180560117703862);
+
+ constant LOGF_TAIL: REAL_VECTOR(0 TO N) := (
+ 0.0,
+ -0.00000000000000543229938420049,
+ 0.00000000000000172745674997061,
+ -0.00000000000001323017818229233,
+ -0.00000000000001154527628289872,
+ -0.00000000000000466529469958300,
+ 0.00000000000005148849572685810,
+ -0.00000000000002532168943117445,
+ -0.00000000000005213620639136504,
+ -0.00000000000001819506003016881,
+ 0.00000000000006329065958724544,
+ 0.00000000000008614512936087814,
+ -0.00000000000007355770219435028,
+ 0.00000000000009638067658552277,
+ 0.00000000000007598636597194141,
+ 0.00000000000002579999128306990,
+ -0.00000000000004654729747598444,
+ -0.00000000000007556920687451336,
+ 0.00000000000010195735223708472,
+ -0.00000000000017319034406422306,
+ -0.00000000000007718001336828098,
+ 0.00000000000010980754099855238,
+ -0.00000000000002047235780046195,
+ -0.00000000000008372091099235912,
+ 0.00000000000014088127937111135,
+ 0.00000000000012869017157588257,
+ 0.00000000000017788850778198106,
+ 0.00000000000006440856150696891,
+ 0.00000000000016132822667240822,
+ -0.00000000000007540916511956188,
+ -0.00000000000000036507188831790,
+ 0.00000000000009120937249914984,
+ 0.00000000000018567570959796010,
+ -0.00000000000003149265065191483,
+ -0.00000000000009309459495196889,
+ 0.00000000000017914338601329117,
+ -0.00000000000001302979717330866,
+ 0.00000000000023097385217586939,
+ 0.00000000000023999540484211737,
+ 0.00000000000015393776174455408,
+ -0.00000000000036870428315837678,
+ 0.00000000000036920375082080089,
+ -0.00000000000009383417223663699,
+ 0.00000000000009433398189512690,
+ 0.00000000000041481318704258568,
+ -0.00000000000003792316480209314,
+ 0.00000000000008403156304792424,
+ -0.00000000000034262934348285429,
+ 0.00000000000043712191957429145,
+ -0.00000000000010475750058776541,
+ -0.00000000000011118671389559323,
+ 0.00000000000037549577257259853,
+ 0.00000000000013912841212197565,
+ 0.00000000000010775743037572640,
+ 0.00000000000029391859187648000,
+ -0.00000000000042790509060060774,
+ 0.00000000000022774076114039555,
+ 0.00000000000010849569622967912,
+ -0.00000000000023073801945705758,
+ 0.00000000000015761203773969435,
+ 0.00000000000003345710269544082,
+ -0.00000000000041525158063436123,
+ 0.00000000000032655698896907146,
+ -0.00000000000044704265010452446,
+ 0.00000000000034527647952039772,
+ -0.00000000000007048962392109746,
+ 0.00000000000011776978751369214,
+ -0.00000000000010774341461609578,
+ 0.00000000000021863343293215910,
+ 0.00000000000024132639491333131,
+ 0.00000000000039057462209830700,
+ -0.00000000000026570679203560751,
+ 0.00000000000037135141919592021,
+ -0.00000000000017166921336082431,
+ -0.00000000000028658285157914353,
+ -0.00000000000023812542263446809,
+ 0.00000000000006576659768580062,
+ -0.00000000000028210143846181267,
+ 0.00000000000010701931762114254,
+ 0.00000000000018119346366441110,
+ 0.00000000000009840465278232627,
+ -0.00000000000033149150282752542,
+ -0.00000000000018302857356041668,
+ -0.00000000000016207400156744949,
+ 0.00000000000048303314949553201,
+ -0.00000000000071560553172382115,
+ 0.00000000000088821239518571855,
+ -0.00000000000030900580513238244,
+ -0.00000000000061076551972851496,
+ 0.00000000000035659969663347830,
+ 0.00000000000035782396591276383,
+ -0.00000000000046226087001544578,
+ 0.00000000000062279762917225156,
+ 0.00000000000072838947272065741,
+ 0.00000000000026809646615211673,
+ -0.00000000000010960825046059278,
+ 0.00000000000002311949383800537,
+ -0.00000000000058469058005299247,
+ -0.00000000000002103748251144494,
+ -0.00000000000023323182945587408,
+ -0.00000000000042333694288141916,
+ -0.00000000000043933937969737844,
+ 0.00000000000041341647073835565,
+ 0.00000000000006841763641591466,
+ 0.00000000000047585534004430641,
+ 0.00000000000083679678674757695,
+ -0.00000000000085763734646658640,
+ 0.00000000000021913281229340092,
+ -0.00000000000062242842536431148,
+ -0.00000000000010983594325438430,
+ 0.00000000000065310431377633651,
+ -0.00000000000047580199021710769,
+ -0.00000000000037854251265457040,
+ 0.00000000000040939233218678664,
+ 0.00000000000087424383914858291,
+ 0.00000000000025218188456842882,
+ -0.00000000000003608131360422557,
+ -0.00000000000050518555924280902,
+ 0.00000000000078699403323355317,
+ -0.00000000000067020876961949060,
+ 0.00000000000016108575753932458,
+ 0.00000000000058527188436251509,
+ -0.00000000000035246757297904791,
+ -0.00000000000018372084495629058,
+ 0.00000000000088606689813494916,
+ 0.00000000000066486268071468700,
+ 0.00000000000063831615170646519,
+ 0.00000000000025144230728376072,
+ -0.00000000000017239444525614834);
+
+ variable M, J:INTEGER;
+ variable F1, F2, G, Q, U, U2, V: REAL;
+ variable ZERO: REAL := 0.0;--Made variable so no constant folding occurs
+ variable ONE: REAL := 1.0; --Made variable so no constant folding occurs
+
+ -- double logb(), ldexp();
+
+ variable U1:REAL;
+
+ begin
+
+ -- Check validity of argument
+ if ( X <= 0.0 ) then
+ assert FALSE
+ report "X <= 0.0 in LOG(X)"
+ severity ERROR;
+ return(REAL'LOW);
+ end if;
+
+ -- Compute value for special cases
+ if ( X = 1.0 ) then
+ return 0.0;
+ end if;
+
+ if ( X = MATH_E ) then
+ return 1.0;
+ end if;
+
+ -- Argument reduction: 1 <= g < 2; x/2^m = g;
+ -- y = F*(1 + f/F) for |f| <= 2^-8
+
+ M := ILOGB(X);
+ G := LDEXP(X, -M);
+ J := INTEGER(REAL(N)*(G-1.0)); -- C code adds 0.5 for rounding
+ F1 := (1.0/REAL(N)) * REAL(J) + 1.0; --F1*128 is an INTEGER in [128,512]
+ F2 := G - F1;
+
+ -- Approximate expansion for log(1+f2/F1) ~= u + q
+ G := 1.0/(2.0*F1+F2);
+ U := 2.0*F2*G;
+ V := U*U;
+ Q := U*V*(A1 + V*(A2 + V*(A3 + V*A4)));
+
+ -- Case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8,
+ -- u1 has at most 35 bits, and F1*u1 is exact, as F1 has < 8 bits.
+ -- It also adds exactly to |m*log2_hi + log_F_head[j] | < 750.
+ --
+ if ( J /= 0 or M /= 0) then
+ U1 := U + 513.0;
+ U1 := U1 - 513.0;
+
+ -- Case 2: |1-x| < 1/256. The m- and j- dependent terms are zero
+ -- u1 = u to 24 bits.
+ --
+ else
+ U1 := U;
+ --TRUNC(U1); --In c this is u1 = (double) (float) (u1)
+ end if;
+
+ U2 := (2.0*(F2 - F1*U1) - U1*F2) * G;
+ -- u1 + u2 = 2f/(2F+f) to extra precision.
+
+ -- log(x) = log(2^m*F1*(1+f2/F1)) =
+ -- (m*log2_hi+LOGF_HEAD(j)+u1) + (m*log2_lo+LOGF_TAIL(j)+q);
+ -- (exact) + (tiny)
+
+ U1 := U1 + REAL(M)*LOGF_HEAD(N) + LOGF_HEAD(J); -- Exact
+ U2 := (U2 + LOGF_TAIL(J)) + Q; -- Tiny
+ U2 := U2 + LOGF_TAIL(N)*REAL(M);
+ return (U1 + U2);
+ end LOG;
+
+
+ function LOG2 (X: in REAL) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) Returns REAL'LOW on error
+ begin
+ -- Check validity of arguments
+ if ( X <= 0.0 ) then
+ assert FALSE
+ report "X <= 0.0 in LOG2(X)"
+ severity ERROR;
+ return(REAL'LOW);
+ end if;
+
+ -- Compute value for special cases
+ if ( X = 1.0 ) then
+ return 0.0;
+ end if;
+
+ if ( X = 2.0 ) then
+ return 1.0;
+ end if;
+
+ -- Compute value for general case
+ return ( MATH_LOG2_OF_E*LOG(X) );
+ end LOG2;
+
+
+ function LOG10 (X: in REAL) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) Returns REAL'LOW on error
+ begin
+ -- Check validity of arguments
+ if ( X <= 0.0 ) then
+ assert FALSE
+ report "X <= 0.0 in LOG10(X)"
+ severity ERROR;
+ return(REAL'LOW);
+ end if;
+
+ -- Compute value for special cases
+ if ( X = 1.0 ) then
+ return 0.0;
+ end if;
+
+ if ( X = 10.0 ) then
+ return 1.0;
+ end if;
+
+ -- Compute value for general case
+ return ( MATH_LOG10_OF_E*LOG(X) );
+ end LOG10;
+
+
+ function LOG (X: in REAL; BASE: in REAL) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) Returns REAL'LOW on error
+ begin
+ -- Check validity of arguments
+ if ( X <= 0.0 ) then
+ assert FALSE
+ report "X <= 0.0 in LOG(X, BASE)"
+ severity ERROR;
+ return(REAL'LOW);
+ end if;
+
+ if ( BASE <= 0.0 or BASE = 1.0 ) then
+ assert FALSE
+ report "BASE <= 0.0 or BASE = 1.0 in LOG(X, BASE)"
+ severity ERROR;
+ return(REAL'LOW);
+ end if;
+
+ -- Compute value for special cases
+ if ( X = 1.0 ) then
+ return 0.0;
+ end if;
+
+ if ( X = BASE ) then
+ return 1.0;
+ end if;
+
+ -- Compute value for general case
+ return ( LOG(X)/LOG(BASE));
+ end LOG;
+
+
+ function SIN (X : in REAL ) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) SIN(-X) = -SIN(X)
+ -- b) SIN(X) = X if ABS(X) < EPS
+ -- c) SIN(X) = X - X**3/3! if EPS < ABS(X) < BASE_EPS
+ -- d) SIN(MATH_PI_OVER_2 - X) = COS(X)
+ -- e) COS(X) = 1.0 - 0.5*X**2 if ABS(X) < EPS
+ -- f) COS(X) = 1.0 - 0.5*X**2 + (X**4)/4! if
+ -- EPS< ABS(X) <BASE_EPS
+
+ constant EPS : REAL := BASE_EPS*BASE_EPS; -- Convergence criteria
+
+ variable N : INTEGER;
+ variable NEGATIVE : BOOLEAN := X < 0.0;
+ variable XLOCAL : REAL := ABS(X) ;
+ variable VALUE: REAL;
+ variable TEMP : REAL;
+
+ begin
+ -- Make XLOCAL < MATH_2_PI
+ if XLOCAL > MATH_2_PI then
+ TEMP := FLOOR(XLOCAL/MATH_2_PI);
+ XLOCAL := XLOCAL - TEMP*MATH_2_PI;
+ end if;
+
+ if XLOCAL < 0.0 then
+ assert FALSE
+ report "XLOCAL <= 0.0 after reduction in SIN(X)"
+ severity ERROR;
+ XLOCAL := -XLOCAL;
+ end if;
+
+ -- Compute value for special cases
+ if XLOCAL = 0.0 or XLOCAL = MATH_2_PI or XLOCAL = MATH_PI then
+ return 0.0;
+ end if;
+
+ if XLOCAL = MATH_PI_OVER_2 then
+ if NEGATIVE then
+ return -1.0;
+ else
+ return 1.0;
+ end if;
+ end if;
+
+ if XLOCAL = MATH_3_PI_OVER_2 then
+ if NEGATIVE then
+ return 1.0;
+ else
+ return -1.0;
+ end if;
+ end if;
+
+ if XLOCAL < EPS then
+ if NEGATIVE then
+ return -XLOCAL;
+ else
+ return XLOCAL;
+ end if;
+ else
+ if XLOCAL < BASE_EPS then
+ TEMP := XLOCAL - (XLOCAL*XLOCAL*XLOCAL)/6.0;
+ if NEGATIVE then
+ return -TEMP;
+ else
+ return TEMP;
+ end if;
+ end if;
+ end if;
+
+ TEMP := MATH_PI - XLOCAL;
+ if ABS(TEMP) < EPS then
+ if NEGATIVE then
+ return -TEMP;
+ else
+ return TEMP;
+ end if;
+ else
+ if ABS(TEMP) < BASE_EPS then
+ TEMP := TEMP - (TEMP*TEMP*TEMP)/6.0;
+ if NEGATIVE then
+ return -TEMP;
+ else
+ return TEMP;
+ end if;
+ end if;
+ end if;
+
+ TEMP := MATH_2_PI - XLOCAL;
+ if ABS(TEMP) < EPS then
+ if NEGATIVE then
+ return TEMP;
+ else
+ return -TEMP;
+ end if;
+ else
+ if ABS(TEMP) < BASE_EPS then
+ TEMP := TEMP - (TEMP*TEMP*TEMP)/6.0;
+ if NEGATIVE then
+ return TEMP;
+ else
+ return -TEMP;
+ end if;
+ end if;
+ end if;
+
+ TEMP := ABS(MATH_PI_OVER_2 - XLOCAL);
+ if TEMP < EPS then
+ TEMP := 1.0 - TEMP*TEMP*0.5;
+ if NEGATIVE then
+ return -TEMP;
+ else
+ return TEMP;
+ end if;
+ else
+ if TEMP < BASE_EPS then
+ TEMP := 1.0 -TEMP*TEMP*0.5 + TEMP*TEMP*TEMP*TEMP/24.0;
+ if NEGATIVE then
+ return -TEMP;
+ else
+ return TEMP;
+ end if;
+ end if;
+ end if;
+
+ TEMP := ABS(MATH_3_PI_OVER_2 - XLOCAL);
+ if TEMP < EPS then
+ TEMP := 1.0 - TEMP*TEMP*0.5;
+ if NEGATIVE then
+ return TEMP;
+ else
+ return -TEMP;
+ end if;
+ else
+ if TEMP < BASE_EPS then
+ TEMP := 1.0 -TEMP*TEMP*0.5 + TEMP*TEMP*TEMP*TEMP/24.0;
+ if NEGATIVE then
+ return TEMP;
+ else
+ return -TEMP;
+ end if;
+ end if;
+ end if;
+
+ -- Compute value for general cases
+ if ((XLOCAL < MATH_PI_OVER_2 ) and (XLOCAL > 0.0)) then
+ VALUE:= CORDIC( KC, 0.0, x, 27, ROTATION)(1);
+ end if;
+
+ N := INTEGER ( FLOOR(XLOCAL/MATH_PI_OVER_2));
+ case QUADRANT( N mod 4) is
+ when 0 =>
+ VALUE := CORDIC( KC, 0.0, XLOCAL, 27, ROTATION)(1);
+ when 1 =>
+ VALUE := CORDIC( KC, 0.0, XLOCAL - MATH_PI_OVER_2, 27,
+ ROTATION)(0);
+ when 2 =>
+ VALUE := -CORDIC( KC, 0.0, XLOCAL - MATH_PI, 27, ROTATION)(1);
+ when 3 =>
+ VALUE := -CORDIC( KC, 0.0, XLOCAL - MATH_3_PI_OVER_2, 27,
+ ROTATION)(0);
+ end case;
+
+ if NEGATIVE then
+ return -VALUE;
+ else
+ return VALUE;
+ end if;
+ end SIN;
+
+
+ function COS (X : in REAL) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) COS(-X) = COS(X)
+ -- b) COS(X) = SIN(MATH_PI_OVER_2 - X)
+ -- c) COS(MATH_PI + X) = -COS(X)
+ -- d) COS(X) = 1.0 - X*X/2.0 if ABS(X) < EPS
+ -- e) COS(X) = 1.0 - 0.5*X**2 + (X**4)/4! if
+ -- EPS< ABS(X) <BASE_EPS
+ --
+ constant EPS : REAL := BASE_EPS*BASE_EPS;
+
+ variable XLOCAL : REAL := ABS(X);
+ variable VALUE: REAL;
+ variable TEMP : REAL;
+
+ begin
+ -- Make XLOCAL < MATH_2_PI
+ if XLOCAL > MATH_2_PI then
+ TEMP := FLOOR(XLOCAL/MATH_2_PI);
+ XLOCAL := XLOCAL - TEMP*MATH_2_PI;
+ end if;
+
+ if XLOCAL < 0.0 then
+ assert FALSE
+ report "XLOCAL <= 0.0 after reduction in COS(X)"
+ severity ERROR;
+ XLOCAL := -XLOCAL;
+ end if;
+
+ -- Compute value for special cases
+ if XLOCAL = 0.0 or XLOCAL = MATH_2_PI then
+ return 1.0;
+ end if;
+
+ if XLOCAL = MATH_PI then
+ return -1.0;
+ end if;
+
+ if XLOCAL = MATH_PI_OVER_2 or XLOCAL = MATH_3_PI_OVER_2 then
+ return 0.0;
+ end if;
+
+ TEMP := ABS(XLOCAL);
+ if ( TEMP < EPS) then
+ return (1.0 - 0.5*TEMP*TEMP);
+ else
+ if (TEMP < BASE_EPS) then
+ return (1.0 -0.5*TEMP*TEMP + TEMP*TEMP*TEMP*TEMP/24.0);
+ end if;
+ end if;
+
+ TEMP := ABS(XLOCAL -MATH_2_PI);
+ if ( TEMP < EPS) then
+ return (1.0 - 0.5*TEMP*TEMP);
+ else
+ if (TEMP < BASE_EPS) then
+ return (1.0 -0.5*TEMP*TEMP + TEMP*TEMP*TEMP*TEMP/24.0);
+ end if;
+ end if;
+
+ TEMP := ABS (XLOCAL - MATH_PI);
+ if TEMP < EPS then
+ return (-1.0 + 0.5*TEMP*TEMP);
+ else
+ if (TEMP < BASE_EPS) then
+ return (-1.0 +0.5*TEMP*TEMP - TEMP*TEMP*TEMP*TEMP/24.0);
+ end if;
+ end if;
+
+ -- Compute value for general cases
+ return SIN(MATH_PI_OVER_2 - XLOCAL);
+ end COS;
+
+ function TAN (X : in REAL) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) TAN(0.0) = 0.0
+ -- b) TAN(-X) = -TAN(X)
+ -- c) Returns REAL'LOW on error if X < 0.0
+ -- d) Returns REAL'HIGH on error if X > 0.0
+
+ variable NEGATIVE : BOOLEAN := X < 0.0;
+ variable XLOCAL : REAL := ABS(X) ;
+ variable VALUE: REAL;
+ variable TEMP : REAL;
+
+ begin
+ -- Make 0.0 <= XLOCAL <= MATH_2_PI
+ if XLOCAL > MATH_2_PI then
+ TEMP := FLOOR(XLOCAL/MATH_2_PI);
+ XLOCAL := XLOCAL - TEMP*MATH_2_PI;
+ end if;
+
+ if XLOCAL < 0.0 then
+ assert FALSE
+ report "XLOCAL <= 0.0 after reduction in TAN(X)"
+ severity ERROR;
+ XLOCAL := -XLOCAL;
+ end if;
+
+ -- Check validity of argument
+ if XLOCAL = MATH_PI_OVER_2 then
+ assert FALSE
+ report "X is a multiple of MATH_PI_OVER_2 in TAN(X)"
+ severity ERROR;
+ if NEGATIVE then
+ return(REAL'LOW);
+ else
+ return(REAL'HIGH);
+ end if;
+ end if;
+
+ if XLOCAL = MATH_3_PI_OVER_2 then
+ assert FALSE
+ report "X is a multiple of MATH_3_PI_OVER_2 in TAN(X)"
+ severity ERROR;
+ if NEGATIVE then
+ return(REAL'HIGH);
+ else
+ return(REAL'LOW);
+ end if;
+ end if;
+
+ -- Compute value for special cases
+ if XLOCAL = 0.0 or XLOCAL = MATH_PI then
+ return 0.0;
+ end if;
+
+ -- Compute value for general cases
+ VALUE := SIN(XLOCAL)/COS(XLOCAL);
+ if NEGATIVE then
+ return -VALUE;
+ else
+ return VALUE;
+ end if;
+ end TAN;
+
+ function ARCSIN (X : in REAL ) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) ARCSIN(-X) = -ARCSIN(X)
+ -- b) Returns X on error
+
+ variable NEGATIVE : BOOLEAN := X < 0.0;
+ variable XLOCAL : REAL := ABS(X);
+ variable VALUE : REAL;
+
+ begin
+ -- Check validity of arguments
+ if XLOCAL > 1.0 then
+ assert FALSE
+ report "ABS(X) > 1.0 in ARCSIN(X)"
+ severity ERROR;
+ return X;
+ end if;
+
+ -- Compute value for special cases
+ if XLOCAL = 0.0 then
+ return 0.0;
+ elsif XLOCAL = 1.0 then
+ if NEGATIVE then
+ return -MATH_PI_OVER_2;
+ else
+ return MATH_PI_OVER_2;
+ end if;
+ end if;
+
+ -- Compute value for general cases
+ if XLOCAL < 0.9 then
+ VALUE := ARCTAN(XLOCAL/(SQRT(1.0 - XLOCAL*XLOCAL)));
+ else
+ VALUE := MATH_PI_OVER_2 - ARCTAN(SQRT(1.0 - XLOCAL*XLOCAL)/XLOCAL);
+ end if;
+
+ if NEGATIVE then
+ VALUE := -VALUE;
+ end if;
+
+ return VALUE;
+ end ARCSIN;
+
+ function ARCCOS (X : in REAL) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) ARCCOS(-X) = MATH_PI - ARCCOS(X)
+ -- b) Returns X on error
+
+ variable NEGATIVE : BOOLEAN := X < 0.0;
+ variable XLOCAL : REAL := ABS(X);
+ variable VALUE : REAL;
+
+ begin
+ -- Check validity of argument
+ if XLOCAL > 1.0 then
+ assert FALSE
+ report "ABS(X) > 1.0 in ARCCOS(X)"
+ severity ERROR;
+ return X;
+ end if;
+
+ -- Compute value for special cases
+ if X = 1.0 then
+ return 0.0;
+ elsif X = 0.0 then
+ return MATH_PI_OVER_2;
+ elsif X = -1.0 then
+ return MATH_PI;
+ end if;
+
+ -- Compute value for general cases
+ if XLOCAL > 0.9 then
+ VALUE := ARCTAN(SQRT(1.0 - XLOCAL*XLOCAL)/XLOCAL);
+ else
+ VALUE := MATH_PI_OVER_2 - ARCTAN(XLOCAL/SQRT(1.0 - XLOCAL*XLOCAL));
+ end if;
+
+
+ if NEGATIVE then
+ VALUE := MATH_PI - VALUE;
+ end if;
+
+ return VALUE;
+ end ARCCOS;
+
+
+ function ARCTAN (Y : in REAL) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) ARCTAN(-Y) = -ARCTAN(Y)
+ -- b) ARCTAN(Y) = -ARCTAN(1.0/Y) + MATH_PI_OVER_2 for |Y| > 1.0
+ -- c) ARCTAN(Y) = Y for |Y| < EPS
+
+ constant EPS : REAL := BASE_EPS*BASE_EPS*BASE_EPS;
+
+ variable NEGATIVE : BOOLEAN := Y < 0.0;
+ variable RECIPROCAL : BOOLEAN;
+ variable YLOCAL : REAL := ABS(Y);
+ variable VALUE : REAL;
+
+ begin
+ -- Make argument |Y| <=1.0
+ if YLOCAL > 1.0 then
+ YLOCAL := 1.0/YLOCAL;
+ RECIPROCAL := TRUE;
+ else
+ RECIPROCAL := FALSE;
+ end if;
+
+ -- Compute value for special cases
+ if YLOCAL = 0.0 then
+ if RECIPROCAL then
+ if NEGATIVE then
+ return (-MATH_PI_OVER_2);
+ else
+ return (MATH_PI_OVER_2);
+ end if;
+ else
+ return 0.0;
+ end if;
+ end if;
+
+ if YLOCAL < EPS then
+ if NEGATIVE then
+ if RECIPROCAL then
+ return (-MATH_PI_OVER_2 + YLOCAL);
+ else
+ return -YLOCAL;
+ end if;
+ else
+ if RECIPROCAL then
+ return (MATH_PI_OVER_2 - YLOCAL);
+ else
+ return YLOCAL;
+ end if;
+ end if;
+ end if;
+
+ -- Compute value for general cases
+ VALUE := CORDIC( 1.0, YLOCAL, 0.0, 27, VECTORING )(2);
+
+ if RECIPROCAL then
+ VALUE := MATH_PI_OVER_2 - VALUE;
+ end if;
+
+ if NEGATIVE then
+ VALUE := -VALUE;
+ end if;
+
+ return VALUE;
+ end ARCTAN;
+
+
+ function ARCTAN (Y : in REAL; X : in REAL) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) Returns 0.0 on error
+
+ variable YLOCAL : REAL;
+ variable VALUE : REAL;
+ begin
+
+ -- Check validity of arguments
+ if (Y = 0.0 and X = 0.0 ) then
+ assert FALSE report
+ "ARCTAN(0.0, 0.0) is undetermined"
+ severity ERROR;
+ return 0.0;
+ end if;
+
+ -- Compute value for special cases
+ if Y = 0.0 then
+ if X > 0.0 then
+ return 0.0;
+ else
+ return MATH_PI;
+ end if;
+ end if;
+
+ if X = 0.0 then
+ if Y > 0.0 then
+ return MATH_PI_OVER_2;
+ else
+ return -MATH_PI_OVER_2;
+ end if;
+ end if;
+
+
+ -- Compute value for general cases
+ YLOCAL := ABS(Y/X);
+
+ VALUE := ARCTAN(YLOCAL);
+
+ if X < 0.0 then
+ VALUE := MATH_PI - VALUE;
+ end if;
+
+ if Y < 0.0 then
+ VALUE := -VALUE;
+ end if;
+
+ return VALUE;
+ end ARCTAN;
+
+
+ function SINH (X : in REAL) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) Returns (EXP(X) - EXP(-X))/2.0
+ -- b) SINH(-X) = SINH(X)
+
+ variable NEGATIVE : BOOLEAN := X < 0.0;
+ variable XLOCAL : REAL := ABS(X);
+ variable TEMP : REAL;
+ variable VALUE : REAL;
+
+ begin
+ -- Compute value for special cases
+ if XLOCAL = 0.0 then
+ return 0.0;
+ end if;
+
+ -- Compute value for general cases
+ TEMP := EXP(XLOCAL);
+ VALUE := (TEMP - 1.0/TEMP)*0.5;
+
+ if NEGATIVE then
+ VALUE := -VALUE;
+ end if;
+
+ return VALUE;
+ end SINH;
+
+ function COSH (X : in REAL) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) Returns (EXP(X) + EXP(-X))/2.0
+ -- b) COSH(-X) = COSH(X)
+
+ variable XLOCAL : REAL := ABS(X);
+ variable TEMP : REAL;
+ variable VALUE : REAL;
+ begin
+ -- Compute value for special cases
+ if XLOCAL = 0.0 then
+ return 1.0;
+ end if;
+
+
+ -- Compute value for general cases
+ TEMP := EXP(XLOCAL);
+ VALUE := (TEMP + 1.0/TEMP)*0.5;
+
+ return VALUE;
+ end COSH;
+
+ function TANH (X : in REAL) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) Returns (EXP(X) - EXP(-X))/(EXP(X) + EXP(-X))
+ -- b) TANH(-X) = -TANH(X)
+
+ variable NEGATIVE : BOOLEAN := X < 0.0;
+ variable XLOCAL : REAL := ABS(X);
+ variable TEMP : REAL;
+ variable VALUE : REAL;
+
+ begin
+ -- Compute value for special cases
+ if XLOCAL = 0.0 then
+ return 0.0;
+ end if;
+
+ -- Compute value for general cases
+ TEMP := EXP(XLOCAL);
+ VALUE := (TEMP - 1.0/TEMP)/(TEMP + 1.0/TEMP);
+
+ if NEGATIVE then
+ return -VALUE;
+ else
+ return VALUE;
+ end if;
+ end TANH;
+
+ function ARCSINH (X : in REAL) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) Returns LOG( X + SQRT( X*X + 1.0))
+
+ begin
+ -- Compute value for special cases
+ if X = 0.0 then
+ return 0.0;
+ end if;
+
+ -- Compute value for general cases
+ return ( LOG( X + SQRT( X*X + 1.0)) );
+ end ARCSINH;
+
+
+
+ function ARCCOSH (X : in REAL) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) Returns LOG( X + SQRT( X*X - 1.0)); X >= 1.0
+ -- b) Returns X on error
+
+ begin
+ -- Check validity of arguments
+ if X < 1.0 then
+ assert FALSE
+ report "X < 1.0 in ARCCOSH(X)"
+ severity ERROR;
+ return X;
+ end if;
+
+ -- Compute value for special cases
+ if X = 1.0 then
+ return 0.0;
+ end if;
+
+ -- Compute value for general cases
+ return ( LOG( X + SQRT( X*X - 1.0)));
+ end ARCCOSH;
+
+ function ARCTANH (X : in REAL) return REAL is
+ -- Description:
+ -- See function declaration in IEEE Std 1076.2-1996
+ -- Notes:
+ -- a) Returns (LOG( (1.0 + X)/(1.0 - X)))/2.0 ; | X | < 1.0
+ -- b) Returns X on error
+ begin
+ -- Check validity of arguments
+ if ABS(X) >= 1.0 then
+ assert FALSE
+ report "ABS(X) >= 1.0 in ARCTANH(X)"
+ severity ERROR;
+ return X;
+ end if;
+
+ -- Compute value for special cases
+ if X = 0.0 then
+ return 0.0;
+ end if;
+
+ -- Compute value for general cases
+ return( 0.5*LOG( (1.0+X)/(1.0-X) ) );
+ end ARCTANH;
+
+end MATH_REAL;
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