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@part[NUMBER, Root "TMAN.MSS"] @Comment{-*-System:TMAN-*-}
@chap[Numbers]
@iix[Numbers] in @Tau[] are objects which represent real numbers.
Numbers come in three varieties: integers, ratios, and floating point
numbers. Integers and ratios are exact models of the corresponding
mathematical objects. Floating point numbers give discrete approximations
to real numbers within a certain implementation-dependent range.
@dc{ Talk about external syntax of numbers. }
As expressions, numbers are self-evaluating literals (see section
@ref[EQ?]).
@begin[ProgramExample]
-377 @ev[] -377
72723460248141 @ev[] 72723460248141
13/5 @ev[] 13/5
34.55 @ev[] 34.55
@end[ProgramExample]
@dc{
(define (fib)
(do ((a 0 b)
(b 1 (+ a b))
(n 1 (+ n 1)))
(nil)
(format t " (fib ~s) = ~s~%" n b)))
(define tau (/ (+ 1 (sqrt 5.0)) 2))
(define (fib n)
(/ (+ (expt tau n) (expt tau (- 0 n))) (sqrt 5.0)))
}
There may be many different objects which all represent the same number.
The effect of this is that @tc[EQ?] may behave in nonintuitive ways when
applied to numbers. It is guaranteed that if two numbers have different
types or different numerical values, then they are @i[not]
@tc[EQ?], but two numbers that appear to be the same may or may not be
@tc[EQ?], even though there is no other way to distinguish them. Use
@tc[EQUAL?] or @tc[=] (page @PageRef(EQUAL?)) for comparing numbers.
@section[Predicates]
@info[NOTES="Type predicate",EQUIV="NUMBERP"]
@desc[@el[](NUMBER? @i[object]) @yl[] @i[boolean]]
Returns true if @i[object] is a number.
@EndDesc[NUMBER?]
@info[NOTES="Type predicate",EQUIV="FIXP"]
@desc[(INTEGER? @i[object]) @yl[] @i[boolean]]
Returns true if @i[object] is an integer.
@EndDesc[INTEGER?]
@info[NOTES="Type predicate",EQUIV="FLOATP"]
@desc[(FLOAT? @i[object]) @yl[] @i[boolean]]
Returns true if @i[object] is a floating point number.
@EndDesc[FLOAT?]
@info[NOTES="Type predicate"]
@desc[(RATIO? @i[object]) @yl[] @i[boolean]]
Returns true if @i[object] is a ratio.
@EndDesc[RATIO?]
@info[EQUIV="ODDP"]
@desc[(ODD? @i[integer]) @yl[] @i[boolean]]
Returns true if @i[integer] is odd.
@EndDesc[ODD?]
@desc[(EVEN? @i[integer]) @yl[] @i[boolean]]
Returns true if @i[integer] is even.
@EndDesc[EVEN?]
@begin[group]
@section[Arithmetic]
@info[EQUIV="PLUS"]
@descN[
F1="@el[](+ . @i[numbers]) @yl[] @i[number]", FN1="+", NL1,
F2="(ADD . @i[numbers]) @yl[] @i[number]", FN2="ADD"
]
Returns the sum of the @i[numbers].
@begin[ProgramExample]
(+ 10 27) @ev[] 37
(+ 10 27.8) @ev[] 37.8
(+) @ev[] 0
(+ -35) @ev[] -35
(+ 1 2 3) @ev[] 6
@end[ProgramExample]
@EndDescN[]
@end[group]
@info[EQUIV="DIFFERENCE"]
@descN[
F1="@el[](- @i[n1 n2]) @yl[] @i[number]", FN1="-" ,
F2="(SUBTRACT @i[n1 n2]) @yl[] @i[number]", FN2="SUBTRACT"
]
Returns the difference of @i[n1] and @i[n2].
@EndDescN[]
@info[EQUIV="MINUS"]
@descN[
F1="@el[](- @i[n]) @yl[] @i[number]",
F2="(NEGATE @i[n]) @yl[] @i[number]", FN2="NEGATE"
]
Returns the additive inverse of @i[n].
@EndDescN[]
@info[EQUIV="TIMES"]
@descN[
F1="@el[](* . @i[numbers]) @yl[] @i[number]", FN1="*" ,
F2="(MULTIPLY . @i[numbers]) @yl[] @i[number]", FN2="MULTIPLY"
]
Returns the product of the @i[numbers].
@EndDescN[]
@info[EQUIV="QUOTIENT"]
@descN[
F1="@el[](/ @i[n1 n2]) @yl[] @i[number]", FN1="/", NL1,
F2="(DIVIDE @i[n1 n2]) @yl[] @i[number]", FN2="DIVIDE"
]
Returns the quotient of @i[n1] and @i[n2].
@begin[ProgramExample]
(/ 20 5) @ev[] 4
(/ 5 20) @ev[] 1/4
(/ 5.0 20) @ev[] 0.25
@end[ProgramExample]
@EndDescN[]
@info[EQUIV="//"]
@desc[(DIV @i[n1 n2]) @yl[] @i[integer]]
Integer division; truncates the quotient of @i[n1] and @i[n2]
towards zero.
@BeginInset[Bug:]
@tc[DIV] in @Timp[] 2.7 does strange things if its arguments
aren't integers.
@EndInset[]
@EndDesc[DIV]
@info[EQUIV="REMAINDER, \"]
@desc[(REMAINDER @i[n1 n2]) @yl[] @i[number]]
Returns the remainder of the division of @i[n1] by @i[n2], with
the sign of @i[n1].
@BeginInset[Bug:]
@tc[REMAINDER] in @Timp[] 2.7 does strange things
if its arguments aren't integers.
@EndInset[]
@EndDesc[REMAINDER]
@dc{
@desc[(FLOOR @i[n1 n2]) @yl[] @i[number]]
Returns the greatest multiple of @i[n2] which is less than or equal to @i[n1];
@qu"round down."
A particularly useful case is where @i[n2] is 1, in which case
@tc[FLOOR] returns @i[n1]'s integer part.
@begin[ProgramExample]
(FLOOR 21 8) @ev[] 16
(FLOOR -21 8) @ev[] -24
(FLOOR 2.67 1) @ev[] 2
@end[ProgramExample]
@EndDesc[FLOOR]
@desc[(CEILING @i[n1 n2]) @yl[] @i[number]]
Returns the least multiple of @i[n2] which is greater than or equal to @i[n1];
@qu"round up."
@begin[ProgramExample]
(CEILING 21 8) @ev[] 24
(CEILING -21 8) @ev[] 16
(CEILING 2.67 1) @ev[] 3
@end[ProgramExample]
@EndDesc[CEILING]
@desc[(TRUNCATE @i[n1 n2]) @yl[] @i[number]]
Returns the greatest multiple of @i[n2] which is less than or equal to the
absolute value of @i[n1], with @i[n1]'s sign;
@qu"round towards zero."
@begin[ProgramExample]
(TRUNCATE 21 8) @ev[] 16
(TRUNCATE -21 8) @ev[] -24
(TRUNCATE 2.67 1) @ev[] 1
@end[ProgramExample]
@EndDesc[TRUNCATE]
@desc[(ROUND @i[n1 n2]) @yl[] @i[number]]
Rounds @i[n1] toward the nearest multiple of @i[n2].
@EndDesc[ROUND]
}
@desc[(MOD @i[n1 n2]) @yl[] @i[number]]
Returns a number @i[k] in the range from zero, inclusive, to @i[n2],
exclusive, such that @i[k] differs from @i[n1] by an integral
multiple of @i[n2]. If @i[n1] is positive, this behaves like
@tc[REMAINDER].
@BeginInset[Bug:]
@tc[MOD] in @Timp[] 2.7 does strange things if its arguments aren't integers.
@EndInset[]
@EndDesc[MOD]
@desc[(EXPT @i[base exponent]) @yl[] @i[number]]
Returns @i[base] raised to the @i[exponent] power.
@i[Base] and @i[exponent] may be any numbers.
@BeginInset[Bug:]
In @Timp[] 2.7, @i[exponent] must be an integer.
@EndInset[]
@EndDesc[EXPT]
@desc[(ABS @i[n]) @yl[] @i[number]]
Returns the absolute value of @i[n].
@EndDesc[ABS]
@desc[(GCD @i[integer1 integer2]) @yl[] @i[integer]]
Returns the greatest common divisor of @i[integer1] and @i[integer2].
@begin[ProgramExample]
(GCD 36 60) @ev[] 12
@end[ProgramExample]
@EndDesc[GCD]
@AnEquivE[Tfn="ADD1",Efn="1+"]
@AnEquivE[Tfn="ADD1",Efn="ADD1"]
@descN[
F1="(ADD1 @i[n]) @yl[] @i[number]", FN1="ADD1",
F2="(1+ @i[n]) @yl[] @i[number]", FN2="1+", NL2
]
Returns @i[n] + 1.
(See also @tc[INCREMENT], page @PageRef[INCREMENT].)
@EndDescN[]
@AnEquivE[Tfn="SUBTRACT1",Efn="1-"]
@AnEquivE[Tfn="SUBTRACT1",Efn="SUB1"]
@descN[
F1="(SUBTRACT1 @i[n]) @yl[] @i[number]", FN1="SUBTRACT1",
F2="(-1+ @i[n]) @yl[] @i[number]", FN2="-1+", NL2
]
Returns @i[n] - 1.
(See also @tc[DECREMENT], page @PageRef[DECREMENT].)
@EndDescN[]
@desc[(MIN . @i[numbers]) @yl[] @i[number]]
Returns the @i[number] with the least numerical magnitude.
(There must be at least one @i[number].)
@EndDesc[MIN]
@desc[(MAX . @i[numbers]) @yl[] @i[number]]
Returns the @i[number] with the greatest numerical magnitude.
(There must be at least one @i[number].)
@EndDesc[MAX]
@section[Comparison]
@descN[
F1="@el[](= @i[n1 n2]) @yl[] @i[boolean]", FN1="=", NL1,
F2="(EQUAL? @i[n1 n2]) @yl[] @i[boolean]", FN2="EQUAL?"
]
@label[EQUAL]
Returns true if @i[n1] is numerically equal to @i[n2].
@EndDescN[]
@AnEquivE[Tfn="LESS?",Efn="LESSP"]
@descN[
F1="@el[](< @i[n1 n2]) @yl[] @i[boolean]", FN1="<", NL1,
F2="(LESS? @i[n1 n2]) @yl[] @i[boolean]", FN2="LESS?"
]
Returns true if @i[n1] is numerically less than @i[n2].
@EndDescN[]
@AnEquivE[Tfn="GREATER?",Efn="GREATERP"]
@descN[
F1="@el[](> @i[n1 n2]) @yl[] @i[boolean]", FN1=">", NL1,
F2="(GREATER? @i[n1 n2]) @yl[] @i[boolean]", FN2="GREATER?"
]
Returns true if @i[n1] is numerically greater than @i[n2].
@EndDescN[]
@descN[
F1="(N= @i[n1 n2]) @yl[] @i[boolean]", FN1="N=", NL1,
F2="(NOT-EQUAL? @i[n1 n2]) @yl[] @i[boolean]", FN2="NOT-EQUAL?"
]
Returns true if @i[n1] is not numerically equal to @i[n2].
@EndDescN[]
@descN[
F1="(>= @i[n1 n2]) @yl[] @i[boolean]", FN1=">=", NL1,
F2="(NOT-LESS? @i[n1 n2]) @yl[] @i[boolean]", FN2="NOT-LESS?"
]
Returns true if @i[n1] is not numerically less than @i[n2].
@EndDescN[]
@descN[
F1="(<= @i[n1 n2]) @yl[] @i[boolean]", FN1="<=", NL1,
F2="(NOT-GREATER? @i[n1 n2]) @yl[] @i[boolean]", FN2="NOT-GREATER?"
]
Returns true if @i[n1] is not numerically greater than @i[n2].
@EndDescN[]
@section[Sign predicates]
@AnEquivE[Tfn="ZERO?",Efn="ZEROP"]
@descN[
F2="(=0? @i[n]) @yl[] @i[boolean]", FN2="=0?", NL2,
F1="(ZERO? @i[n]) @yl[] @i[boolean]", FN1="ZERO?"
]
Returns true if @i[n] is numerically equal to zero.
@EndDescN[]
@AnEquivE[Tfn="NEGATIVE?",Efn="MINUSP"]
@descN[
F2="(<0? @i[n]) @yl[] @i[boolean]", FN2="<0?", NL2,
F1="(NEGATIVE? @i[n]) @yl[] @i[boolean]", FN1="NEGATIVE?"
]
Returns true if @i[n] is negative.
@EndDescN[]
@AnEquivE[Tfn="POSITIVE?",Efn="PLUSP"]
@descN[
F2="(>0? @i[n]) @yl[] @i[boolean]", FN2=">0?", NL2,
F1="(POSITIVE? @i[n]) @yl[] @i[boolean]", FN1="POSITIVE?"
]
Returns true if @i[n] is positive.
@EndDescN[]
@descN[
F2="(N=0? @i[n]) @yl[] @i[boolean]", FN2="N=0?", NL2,
F1="(NOT-ZERO? @i[n]) @yl[] @i[boolean]", FN1="NOT-ZERO?"
]
Returns true if @i[n] is not numerically equal to zero.
@EndDescN[]
@descN[
F2="(>=0? @i[n]) @yl[] @i[boolean]", FN2=">=0?", NL2,
F1="(NOT-NEGATIVE? @i[n]) @yl[] @i[boolean]", FN1="NOT-NEGATIVE?"
]
Returns true if @i[n] is non-negative.
@EndDescN[]
@descN[
F2="(<=0? @i[n]) @yl[] @i[boolean]", FN2="<=0?", NL2,
F1="(NOT-POSITIVE? @i[n]) @yl[] @i[boolean]", FN1="NOT-POSITIVE?"
]
Returns true if @i[n] is non-positive.
@EndDescN[]
@section[Transcendental functions]
@desc[(EXP @i[float]) @yl[] @i[float]]
Exponential function (e@+[@i[x]]).
@EndDesc[EXP]
@desc[(LOG @i[float]) @yl[] @i[float]]
Natural logarithm (log@-[e]@i[x]).
@EndDesc[LOG]
@desc[(SQRT @i[float]) @yl[] @i[float]]
Returns the square root of its argument.
@EndDesc[SQRT]
@desc[(COS @i[float]) @yl[] @i[float]]
Returns the cosine of its argument.
@EndDesc[COS]
@desc[(SIN @i[float]) @yl[] @i[float]]
Returns the sine of its argument.
@EndDesc[SIN]
@desc[(TAN @i[float]) @yl[] @i[float]]
Returns the tangent of its argument.
@EndDesc[TAN]
@desc[(ACOS @i[float1]) @yl[] @i[float2]]
Arccosine. Returns a number whose cosine is @i[float1].
@EndDesc[ACOS]
@desc[(ASIN @i[float1]) @yl[] @i[float2]]
Arcsine. Returns a number whose sine is @i[float1].
@EndDesc[ASIN]
@desc[(ATAN2 @i[x y]) @yl[] @i[float]]
Two-argument arctangent. This computes the
angle between the positive x-axis and the point
(@i[x], @i[y]). For example, if (@i[x], @i[y]) is in the first or third
quadrant, @tc[(ATAN2 @i[x y])] returns the arctangent of @i[y]/@i[x].
@BeginInset[Bug:]
@tc[ATAN2] isn't implemented in @Timp[] 2.7. @Timp[] 2.7 does
implement @tc[ATAN] (arctangent), however.
@EndInset[]
@EndDesc[ATAN2]
@section[Bitwise logical operators]
@BeginInset[Bug:]
In @Timp[] 2.7, the routines in this section are restricted to take
fixnums (see page @pageref[FIXNUM?]), not arbitrary integers.
@EndInset[]
@desc[(LOGAND @i[integer1 integer2]) @yl[] @i[integer]]
Returns the bitwise logical @i[and] of its arguments.
@begin[ProgramExample]
(LOGAND 10 12) @ev[] 8
@end[ProgramExample]
@EndDesc[LOGAND]
@desc[(LOGIOR @i[integer1 integer2]) @yl[] @i[integer]]
Returns the bitwise logical inclusive @i[or] of its arguments.
@EndDesc[LOGIOR]
@desc[(LOGXOR @i[integer1 integer2]) @yl[] @i[integer]]
Returns the bitwise logical exclusive @i[or] of its arguments.
@EndDesc[LOGXOR]
@desc[(LOGNOT @i[integer]) @yl[] @i[integer]]
Returns the bitwise logical @i[not] of its argument.
@EndDesc[LOGNOT]
@desc[(ASH @i[integer amount]) @yl[] @i[integer]]
Shifts @i[integer] to the left @i[amount] bit positions.
If @i[amount] is negative, shifts @i[integer] right by the absolute
value of @i[amount] bit positions.
@EndDesc[ASH]
@desc[(BIT-FIELD @i[integer position size]) @yl[] @i[integer]]
Extracts a bit field out of @i[integer]. The field extracted begins
at bit position @i[position] from the low-order bit of the integer
and consists of @i[size] bits. The extracted field is returned
as a positive integer.
@begin[ProgramExample]
(BIT-FIELD 27 1 3) @ev[] 5
(BIT-FIELD -1 4 5) @ev[] 31
@end[ProgramExample]
@EndDesc[BIT-FIELD]
@desc[(SET-BIT-FIELD @i[integer position size value]) @yl[] @i[integer]]
Inserts a value into a bit field of @i[integer]. The field begins
at bit position @i[position] from the low-order bit of the integer
and consists of @i[size] bits. A new integer is returned which is the same as the
argument @i[integer] except that this field has been altered to be @i[value].
@begin[ProgramExample]
(SET-BIT-FIELD 32 1 3 5) @ev[] 42
@end[ProgramExample]
@EndDesc[SET-BIT-FIELD]
@section[Coercion]
@desc[(->INTEGER @i[number]) @yl[] @i[integer]]
Coerces @i[number] to an integer, truncating towards zero if necessary.
@begin[ProgramExample]
(->INTEGER 17) @ev[] 17
(->INTEGER 7/4) @ev[] 1
(->INTEGER 17.6) @ev[] 17
(->INTEGER -17.6) @ev[] -17
@end[ProgramExample]
@enddesc[->INTEGER]
@desc[(->FLOAT @i[number]) @yl[] @i[float]]
Coerces @i[number] to a floating point number.
@begin[ProgramExample]
(->FLOAT 17) @ev[] 17.0
(->FLOAT 7/4) @ev[] 1.75
(->FLOAT 17.6) @ev[] 17.6
@end[ProgramExample]
@enddesc[->FLOAT]
@section[Assignment]
@tc[INCREMENT] and @tc[DECREMENT] are assignment forms, like @tc[SET]
and @tc[PUSH]. See section @ref[Assignmentsection] for a general
discussion.
@info[Notes "Special form"]
@desc[(INCREMENT @i[location]) @yl[] @i[number]]
Adds one to the value in @i[location], stores the sum back into @i[location],
and yields the sum.
@begin[ProgramExample]
(LSET L 6)
(INCREMENT L) @ev[] 7
L @ev[] 7
(LSET M (LIST 10 20 30)) @ev[] (10 20 30)
(INCREMENT (CAR M)) @ev[] 11
M @ev[] (11 20 30)
(INCREMENT @i[location]) @ce[] (MODIFY @i[location] ADD1)
@end[ProgramExample]
@EndDesc[INCREMENT]
@info[Notes "Special form"]
@desc[(DECREMENT @i[location]) @yl[] @i[number]]
Subtracts one from the value in @i[location], stores the
difference back into @i[location], and yields the difference.
@begin[ProgramExample]
(DECREMENT @i[location]) @ce[] (MODIFY @i[location] SUBTRACT1)
@end[ProgramExample]
@EndDesc[DECREMENT]
