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------------------------------------------------------------------------
--
-- 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,
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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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