------------------------------------------------------------------------
--
-- 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;