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HELP MATH A. Sloman March 1987 Updated Dec 1990 MATHEMATICAL OPERATIONS AND PROCEDURES IN POP-11 Full details are provided in REF * NUMBERS. REF * DATA describes the data-types available in POPLOG, including various types of numbers. This much shorter overview file summarises available facilities. In many cases the command: <ENTER> ?? <name> can be used to obtain more information about the particular procedure or variable mentioned below without explanation. This is normally quicker than searching through REF * NUMBERS CONTENTS - (Use <ENTER> g to access required sections) -- Types of numbers available in POP-11 -- Recogniser procedures -- Arithmetical operators -- Comparison procedures (predicates) -- Operations on Ratios -- Operations on Complex numbers -- Bitwise operations -- Floating point utilities -- Trigonometric and other utilities -- Representation and efficiency -- Fast integer procedures -- Global variables and constants -- LIB FLOAT_PARAMETERS -- LIB INT_PARAMETERS -- See also -- Types of numbers available in POP-11 ------------------------------- BIGINTEGER Arbitrary-precision integer e.g. 123456789123456789123456789 COMPLEX Complex number e.g. 4.5_+:9.94987 (<real>_+:<imaginary>) DDECIMAL Double Floating Point e.g. 99.6123456789 DECIMAL Single Floating Point (one machine long word) e.g. 99.6123 INTEGER Integer (one machine long word) e.g. 123456789 RATIO Ratio of two integers e.g. 3_/4 Integers and decimals are "simple" items, recognised by issimple. All others are compound and are represented by records in the POPLOG heap area. See REF * ITEMISE for information on input formats for numbers. REF * DATA for information on other data types REF * NUMBERS for full information on available numeric operations HELP *EFFICIENCY for tips on optimising programs using numbers -- Recogniser procedures ---------------------------------------------- Recogniser procedures are available, each taking an item and returning a boolean result: isinteger isbiginteger isintegral isratio isrational isdecimal isddecimal isreal iscomplex isnumber Also: issimple iscompound -- Arithmetical operators -------------------------------------------- A range of infix operators is provided for manipulating all the above types of numbers, e.g. X + Y add two numbers X - Y subtract two numbers - X negate one number X * Y multiply two numbers X ** Y raise X to the power Y X / Y divide first number by second (SEE Note below) X // Y divide X by Y and return remainder and quotient. X div Y returns just the quotient. X rem Y returns the remainder on dividing X by Y X mod Y returns the modulus The difference between REM and MOD is that REM always returns a result with the same sign as X while MOD returns a result with the same sign as Y. See HELP * MOD, HELP * REM. -- Note on "/" and ratios --------------------------------------------- Since the introduction of ratios the divide operator "/" has been capable of returning a ratio when given two integers. E.g. 12 / 3 => ** 4 10 / 3 => ** 10_/3 You can prevent the production of a ratio by making one of the arguments a decimal number 10 / 3.0 => ** 3.33333 Whether the result is a single precision decimal or double precision ddecimal will depend on whether POPDPRECISION is TRUE or FALSE. (See Global variables and constants, below). Alternatively you can cause ratios to print out like decimals by setting pop_pr_ratios FALSE: false -> pop_pr_ratios; 10 / 3 => ** 3.333333 The result is still a ratio: only the printing format is altered. Programs which use of ratios or ddecimals can generate more garbage collections than expected because these numbers are compound entities, i.e allocated on the heap. See Representation and efficiency, below. -- Comparison procedures (predicates) --------------------------------- X > Y TRUE if X greater than Y X < Y TRUE if X less than Y X >= Y TRUE if X greater than or equal to Y X <= Y TRUE if X less than or equal to Y X == Y TRUE if X is identical with Y X /== Y TRUE if X is not identical with Y X = Y TRUE if X is equal in value to Y X /= Y TRUE if X is not equal in value to Y X ==# Y TRUE if X == Y or the numbers are of the same type and numeric value. See REF * NUMBERS for the differences btween "=", "==#" and "==" WARNING: comparisons between numbers can give counter-intuitive results if either or both are decimals or ddecimals. Read HELP * EQUAL for more information. The precise definitions are in REF * NUMBERS/Comparisons -- Operations on Ratios ----------------------------------------------- destratio(RAT) -> DENOMINATOR -> NUMERATOR numerator(RAT) -> NUMERATOR denominator(RAT) -> DENOMINATOR -- Operations on Complex numbers -------------------------------------- (For full details see REF * NUMBERS) NUM1 +: NUM2 -> NUM3 NUM1 -: NUM2 -> NUM3 +: NUM1 -> NUM2 -: NUM1 -> NUM2 unary_+:(NUM1) -> NUM2 unary_-:(NUM1) -> NUM2 conjugate(NUM1) -> NUM2 destcomplex(NUM) -> IMAGPART -> REALPART realpart(NUM) -> REALPART imagpart(NUM) -> IMAGPART -- Bitwise operations ------------------------------------------------- INT1 && INT2 -> INT3 - logical and INT1 &&~~ INT2 -> INT3 - logical and_not INT1 || INT2 -> INT3 - logical or INT1 ||/& INT2 -> INT3 - logical exclusive or ~~ INT1 -> INT2 - logical complement INT1 << N -> INT2 - shift INT1 left N bits INT1 >> N -> INT2 - shift INT1 right N bits INT1 &&/=_0 INT2 -> BOOL - logical and and test /== 0 INT1 &&=_0 INT2 -> BOOL - logical and and test == 0 testbit(INT, N) -> BOOL - test Nth bit of INT BOOL -> testbit(INT, N) -> NEWINT - set Nth bit of INT integer_leastbit(INT) -> N - return position of lowest bit integer_length(INT) -> N - return position of highest bit integer_bitcount(INT) -> N - count 1 bits if INT positive, or 0 bits if INT negative integer_field(SIZE, POSITION) -> ACCESS_PDR - create accessing and updating procedures for sub-bitfields -- Floating point utilities ------------------------------------------- These enable machine independent procedures for manipulating floating point numbers to be created. float_decode(FLOAT, INT_MANTISSA) -> FLOAT_SIGN -> INT_EXPO -> MANTISSA float_scale(FLOAT1, INT_EXPO) -> FLOAT2 float_sign(FLOAT_SIGN, FLOAT1) -> FLOAT2 float_digits(FLOAT) -> DIGITS float_precision(FLOAT) -> SIGDIGITS See REF * NUMMBERS for full details. -- Trigonometric and other utilities ---------------------------------- The following are provided. See REF * NUMBERS for full descriptions. abs arccos arccosh arcsin arcsinh arctan arctan2 arctanh cis cos cosh exp fracof gcd_n intof lcm_n log log10 max min negate number_coerce phase random random0 round sign sin sinh sqrt tan tanh Further information is available in REF * NUMBERS. A summary account of any procedure or operation can be obtained by means of the <ENTER> ?? <name> command. Alternatively put the cursor to the left of the name and simply type <ENTER> ?? Some of these have individual neop files giving more information, e.g. HELP * INTOF, * FRACOF -- Representation and efficiency -------------------------------------- Integers and Decimals occupy one machine longword (32 bits), though 2 bits are used for type information, so only 30 bits are available (on 32 bit processors) for the number. These are the only "simple" data-types in POP-11. All the other number types are "compount" and require the construction of a record taking two or more words. Consequently operations producing them can cause garbage collections. Ratios especially can be constructed inadvertently by using "/" with two integer arguments, and this can lead to excessive garbage collection. For tips on avoiding this see HELP * EFFICIENCY -- Fast integer procedures -------------------------------------------- Since the arithmetical operations are defined over all the numeric data-types, they often have to check the nature of their arguments at run time. In order to speed up processing of integers there are several operations which assume their arguments are integers (not bigintegers) and do not do any checking. They are listed in REF * FASTPROCS. Since operations on decimals are slower than on integers it is sometimes useful to employ integers instead of decimals, then transform the results back to decimals at the end. -- Global variables and constants ------------------------------------- pi - (constant) best ddecimal approximation to "pi" pop_float_radix - radix of floating point representation. Currently = 2. popdprecision - controls default conversion of ddecimals to decimals. Default is false, so results of arithmetic operators are decimals. popradians - controls whether trigonometric functions use radians or degrees. pop_pr_ratios - controls whether ratios print using "_/" or decimal notation. ranseed - controls, and is changed by, random. See HELP * RANDOM -- LIB FLOAT_PARAMETERS ----------------------------------------------- This library makes available a range of global implementation specific constants contained in the structure pop_float_parameters pop_most_positive_decimal pop_least_positive_decimal pop_least_negative_decimal pop_most_negative_decimal pop_most_positive_ddecimal pop_least_positive_ddecimal pop_least_negative_ddecimal pop_most_negative_ddecimal pop_plus_epsilon_decimal pop_plus_epsilon_ddecimal These two are the smallest positive numbers in each format which when added to 1.0 of the same format produce a value not equal to 1.0. pop_minus_epsilon_decimal pop_minus_epsilon_ddecimal -- LIB INT_PARAMETERS ----------------------------------------------- This library defines the following parameter constants, whose values relate to the particular implementation of POPLOG in use: pop_max_int The largest (i.e. most positive) integer value represented in immediate format (as an integer) pop_min_int The smallest (i.e. most negative) integer value represented in immediate format (as an integer) -- See also ----------------------------------------------------------- REF * DATA describes range of data types available in POP-11. HELP * NUMBERS describes number data types in POP-11 HELP * EQUAL describes equailty tests in POP-11 HELP * INTOF HELP * FRACOF REF * NUMBERS describes available arithmetic procedures REF * ITEMISE gives information about input formats HELP * RANDOM describes available random number generators REF * INTVEC describes an integer vector type HELP * EFFICIENCY gives information on efficiency issues REF * FASTPROCS fast, non-checking, procedures HELP * PRNUM for printing integers and decimals HELP * PRINTF more general formatted printing HELP * FORMAT_PRINT describes a large number of facilities for formatted printing. --- C.all/help/math ---------------------------------------------------- --- Copyright University of Sussex 1991. All rights reserved. ----------