*line 1: Unknown output type = >$A include=template_file IMP Core Environment Standard Section 3: Mathematical Procedures The procedures described in this section give the IMP programmer access to a range of basic mathematical functions. This range is not intended to be as complete as, for example, the FORTRAN intrinsic library or one of the major mathematical libraries such as NAGLIB. Instead, it is hoped that the needs of the majority of portable applications will be met here without placing an undue burden on the implementor. At present, this standard does not prescribe a level of accuracy which the procedures described here must attain in order for an implementation to conform to the standard. Implementors are instead referred to [CODY80] for an indication of the level of accuracy which can be obtained with some care in implementation. In an ideal world, all mathematical functions would return results which were precisely correct in all cases. In practice, the results of IMP procedures are constrained to be of at most the precision of a long ____ real variable, which means that they will differ from the ideal in some ____ measure for almost all arguments. The accumulation of similar errors during the passing of arguments and computation of results also adds to the overall deviation from the ideal. The procedures here are described for clarity in terms of the ideal: a floating-point system of infinite precision and near-infinite range, and any mathematical relationships or example results given should be interpreted under this model. However, it should always be borne in mind that, unless otherwise stated in the text, any procedure involving real numbers either as arguments, intermediate values or results will be subject to errors of precision for certain argument and result values. 3.1 real to integer Conversion ____ _______ For real to integer conversions, there are two basic operations which ____ _______ are commonly performed. Firstly, the most common is to round a real ____ value to the nearest integer value: this facility is provided by the _______ functions ROUND and INT. Secondly, the programmer may wish to truncate the real value: this operation is embodied in TRUNC and INTPT as ____ described below. The reason for defining a pair of procedures for each of these operations is to cater for the two common definitions of the terms "round" and "truncate": the procedures ROUND and TRUNC are defined in the sense of the national and international standards [BS6192, ISO7185] for the programming language PASCAL, i.e. with truncation being towards zero. These functions are to be preferred for most programs as their operation is easily understood textually, for example ROUND and TRUNC of "-3.6" are "-4" and "-3" respectively. These procedures are referred to 3-1 below as the "textual" versions of the conversion functions. The procedures INT and INTPT are based on an alternative definition of truncation used in certain mathematical contexts. Here, INTPT corresponds to TRUNC except that any truncation required is guaranteed to make the number less positive, i.e. INTPT(X)<=X. This is in contrast to TRUNC's definition which implies that TRUNC(X) will become closer to zero if X is not already integer. Use of INT and INTPT is only recommended after careful consideration of a program's requirements: the results from INTPT in particular can sometimes be unexpected. For example, INTPT(-3.4)=-4 but INTPT(3.4)=3. These procedures are most useful when some mathematical statement is to be made about a program: their effect of modifying all numbers in the same direction on the number line makes them easier to include in such statements. INT and INTPT are referred to as the "monotonic" versions of the conversion functions. In order to give the reader an overall idea of the intent of the procedures defined formally in the two sub-sections which follow, their effect on a range of key values are summarised in the following table. Note that the effects on these key values are duplicated as examples in the definitions of each individual procedure. 'X' Int(X) Int Pt(X) Round(X) Trunc(X) -11.7 -12 -12 -12 -11 -1.2 -1 -2 -1 -1 -0.5 0 -1 -1 0 0.5 1 0 1 0 1.2 1 1 1 1 11.7 12 11 12 11 3.1.1 Textual Versions * integer function TRUNC ( long real X ) _______ ________ ____ ____ This function converts the given long real value into an ____ ____ integer, with any truncation being towards zero (compare with INTPT where the truncation is towards minus infinity). It is identical to the TRUNC function defined in [BS6192] section 6.6.6.3. The TRUNC function obeys the following relationships: X-1 < TRUNC(X) <= X {if X >= 0} X <= TRUNC(X) < X+1 {if X < 0} Examples: TRUNC(-11.7) = -11 TRUNC( -1.2) = -1 TRUNC( -0.5) = 0 TRUNC( 0.5) = 0 TRUNC( 1.2) = 1 TRUNC( 11.7) = 11 It is an error (ERR0007; integer range exceeded) if the value of _______ TRUNC(X) exceeds the implementation defined range for the 3-2 integer data type (DEF0005; range of integer variables). _______ _______ * integer function ROUND ( long real X ) _______ ________ ____ ____ This function returns the integer closest to a given long ____ real value. In the case where the value is exactly halfway ____ between two integer values the value furthest from zero will be returned. It is identical to the ROUND function defined in [BS6192] section 6.6.6.3. The ROUND function obeys the following relations: X-1/2 < ROUND(X) <= X+1/2 {if X >= 0} X-1/2 <= ROUND(X) < X+1/2 {if X < 0} It may also be defined in terms of the TRUNC function by means of the following relationships: ROUND(X) = TRUNC(X+1/2) {if X >= 0} ROUND(X) = TRUNC(X-1/2) {if X < 0} Examples: ROUND(-11.7) = -12 ROUND( -1.2) = -1 ROUND( -0.5) = -1 ROUND( 0.5) = 1 ROUND( 1.2) = 1 ROUND( 11.7) = 12 It is an error (ERR0007; integer range exceeded) if the value of _______ ROUND(X) exceeds the implementation defined range for the integer data type (DEF0005; range of integer variables). _______ _______ 3.1.2 Monotonic Versions * integer function INTPT ( long real X ) _______ ________ ____ ____ This function converts the given long real value into an ____ ____ integer, with any truncation being towards minus infinity (compare with TRUNC where the truncation is towards zero). The INTPT function obeys the following relation: X-1 < INTPT(X) <= X Examples: INTPT(-11.7) = -12 INTPT( -1.2) = -2 INTPT( -0.5) = -1 INTPT( 0.5) = 0 INTPT( 1.2) = 1 INTPT( 11.7) = 11 It is an error (ERR0007; integer range exceeded) if the value of _______ INTPT(X) exceeds the implementation defined range for the 3-3 integer data type (DEF0005; range of integer variables). _______ _______ * integer function INT ( long real X ) _______ ________ ____ ____ This function returns the integer closest to a given long ____ real value. In the case where the argument lies exactly halfway ____ between two integer values the more positive integer value will be returned. The INT function obeys the following relation: X-1/2 < INT(X) <= X+1/2 Alternatively, INT may be related to INTPT as follows: INT(X) = INTPT(X+1/2) Examples: INT(-11.7) = -12 INT( -1.2) = -1 INT( -0.5) = 0 INT( 0.5) = 1 INT( 1.2) = 1 INT( 11.7) = 12 It is an error (ERR0007; integer range exceeded) if the value of _______ INT(X) exceeds the implementation defined range for the integer _______ data type (DEF0005; range of integer variables). _______ 3.2 Trigonometric Functions All angles are in radians. For SIN, COS and TAN, X is not ___ restricted to be less than 2*PI. * constant long real PI ________ ____ ____ The value of the mathematical constant 'pi' expressed to the maximum accuracy of a long real. ____ ____ * long real function SIN ( long real X ) ____ ____ ________ ____ ____ Sine of X * long real function COS ( long real X ) ____ ____ ________ ____ ____ Cosine of X 3-4 * long real function TAN ( long real X ) ____ ____ ________ ____ ____ Tangent of X * long real function ARC SIN ( long real X ) ____ ____ ________ ____ ____ Arc Sine of X. It is an error unless |X| <= 1. The range of the result is -PI/2 <= result <= PI/2. * long real function ARC COS ( long real X ) ____ ____ ________ ____ ____ Arc Cosine of X. It is an error unless |X| <= 1. The range of the result is 0 <= result <= PI. * long real function ARC TAN 1 ( long real X ) ____ ____ ________ ____ ____ Arc Tangent of X. Mathematically, the result range should be -PI/2 < result < PI/2, but the limited precision of machine arithmetic may cause the range to increase to include the two end-points. This is because, at the limits of the argument range (positive and negative numbers approaching the limits of the machine's floating-point range) the mathematically correct result is indistinguishable from +/- PI/2, and is in addition closer to +/- PI/2 than it is to the machine representation of any other number within the exclusive range. In practice, then, the result range of ARC TAN is -PI/2 <= result <= PI/2. * long real function ARC TAN ( long real X, Y ) ____ ____ ________ ____ ____ Arctangent of (Y/X). If Y is positive, the result is positive. If Y is zero, the result is zero if X is positive and PI if X is negative. If Y is negative, the result is negative. If X is zero, the absolute value of the result is PI/2. It is an error if X=Y=0. The range of the result for ARC TAN is: -PI < result <= PI. 3.3 Miscellaneous 3-5 * long real function FRACTION ( long real X ) ____ ____ ________ ____ ____ This function returns the remainder after the parameter has been converted to an integer with truncation towards zero. It _______ obeys the following relation: FRACTION(X) = X-TRUNC(X) so that X = TRUNC(X)+FRACTION(X) Examples: FRACTION(-11.7) = -0.7 FRACTION( -1.2) = -0.2 FRACTION( -0.5) = -0.5 FRACTION( 0.5) = 0.5 FRACTION( 1.2) = 0.2 FRACTION( 11.7) = 0.7 * long real function FRACPT ( long real X ) ____ ____ ________ ____ ____ This function returns the remainder after the parameter has been converted to an integer with truncation towards minus _______ infinity. It obeys the following relation: FRACPT(X) = X-INTPT(X) so that X = INTPT(X)+FRACPT(X) Examples: FRACPT(-11.7) = 0.3 FRACPT( -1.2) = 0.8 FRACPT( -0.5) = 0.5 FRACPT( 0.5) = 0.5 FRACPT( 1.2) = 0.2 FRACPT( 11.7) = 0.7 Note that, because of this definition, FRACPT(X) will always be positive. * long real function FLOAT ( long real X ) ____ ____ ________ ____ ____ This function simply returns its long real parameter as ____ ____ result. Its principal use is in forcing calculations to be performed as real where the compiler might otherwise perform them as integer and risk an integer overflow condition. Example: integer I, J ; real R _______ ____ I = 1000000 ; J = 1000000 R = I*J {may overflow} can become R = FLOAT(I)*FLOAT(J) {usually larger range} FLOAT may also be used to ensure that a particularly critical computation is performed at the higher level of precision offered by long real values. ____ ____ Example: real R, A, B, C ____ A = 1@10 ; B = 1@10 ; C = 1 3-6 Example: R = (A+C)-B {may lose precision} can become R = (FLOAT(A)+C)-FLOAT(B) {usually more precision} * integer function REM ( integer A, B ) _______ ________ _______ This function gives the remainder after A is divided by B in integer arithmetic. It obeys the following relation: REM(A, B) = A - TRUNC(A/B) * B This may alternatively be specified using IMP integer arithmetic _______ as follows: REM(A, B) = A - A//B * B It can be shown that the sign of the result is the same as the sign of the dividend (A). It is an error (ERR0002; Division by zero) if B is zero. Examples: REM(10,10) = 0 REM(10,3) = 1 REM(10,-3) = 1 REM(-10,3) = -1 * integer function MUL DIV ( integer A, B, C ) _______ ________ _______ {informal & provisional} Result is ROUND(A*B/C) to infinite precision and range. * long real function LOG ( long real X ) ____ ____ ________ ____ ____ {informal & provisional} Natural Logarithm of X; Log[base e](X). Error if (X<=0). * long real function EXP ( long real X ) ____ ____ ________ ____ ____ {informal & provisional} Exponential function; e^X. * long real function SQRT ( long real X ) ____ ____ ________ ____ ____ {informal & provisional} Square root; X^(1/2). Error if End of input file Program stopped