4.1cBSD/usr/src/ucb/lisp/doc/ch3.n

." $Header: /na/franz/doc/RCS/ch3.n,v 1.1 83/01/31 07:07:34 jkf Exp $
.Lc Arithmetic\ Functions 3
.pp
This chapter describes 
.Fr "'s"
functions for doing arithmetic.
Often the same function is know by many names, such as
.i add
which is also
.i plus ,
.i sum ,
and \(pl.
This is due to our desire to be compatible with other Lisps.
The 
.Fr
user is advised to avoid using functions with names
such as \(pl and \(** unless
their arguments are fixnums.
The lisp compiler takes advantage of the fact that their
arguments are fixnums.
.pp
An attempt to divide or to generate a floating
point result outside of the range of
floating point numbers
will cause a floating exception signal
from the UNIX operating system.
The user can catch and process this interrupt if he wishes (see the 
description of the
.i signal
function).
.sh 2 simple\ arithmetic\ functions \n(ch 1
.Lf add "['n_arg1 ...]"
.Lx plus "['n_arg1 ...]"
.Lx sum "['n_arg1 ...]"
.Lx \(pl "['x_arg1 ...]"
.Re
the sum of the arguments. If no arguments are given, 0 is returned.
.No
if the size of the partial sum exceeds the limit of a fixnum, the
partial sum will be converted to a bignum.
If any of the arguments are flonums, the partial sum will be 
converted to a flonum when that argument is processed and the 
result will thus be a flonum.
Currently, if in the process of doing the 
addition a bignum must be converted into
a flonum an error message will result.
.Lf add1 'n_arg
.Lx 1+ 'x_arg
.Re
its argument plus 1.
.Lf diff "['n_arg1 ... ]"
.Lx difference "['n_arg1 ... ]"
.Lx \(mi  "['x_arg1 ... ]"
.Re
the result of subtracting from n_arg1 all subsequent arguments. 
If no arguments are given, 0 is returned.
.No
See the description of add for details on data type conversions and
restrictions.
.Lf sub1 "'n_arg"
.Lx 1\(mi "'x_arg"
.Re
its argument minus 1.
.Lf minus "'n_arg"
.Re
zero minus n_arg.
.Lf product "['n_arg1 ... ]"
.Lx times "['n_arg1 ... ]"
.Lx \(** "['x_arg1 ... ]"
.Re
the product of all of its arguments.
It returns 1 if there are no arguments.
.No
See the description of the function \fIadd\fP for details and restrictions to the
automatic data type coercion.
.Lf quotient "['n_arg1 ...]"
.Lx / "['x_arg1 ...]"
.Re
the result of dividing the first argument by succeeding ones.
.No
If there are no arguments, 1 is returned.
See the description of the function \fIadd\fP for details and restrictions
of data type coercion.
A divide by zero will cause a floating exception interrupt -- see 
the description of the
.i signal 
function.
.Lf *quo "'i_x 'i_y"
.Re
the integer part of i_x / i_y.
.Lf Divide "'i_dividend 'i_divisor"
.Re
a list whose car is the quotient and whose cadr is the remainder of the
division of i_dividend by i_divisor.
.No
this is restricted to integer division.
.Lf Emuldiv "'x_fact1 'x_fact2 'x_addn 'x_divisor"
.Re
a list of the quotient and remainder of this operation: 
((x_fact1\ *\ x_fact2)\ +\ (sign\ extended)\ x_addn)\ /\ x_divisor.
.No
this is useful for creating a bignum arithmetic package in Lisp.
.sh 2 predicates
.Lf numberp "'g_arg"
.Lf numbp "'g_arg"
.Re
t iff g_arg is a number (fixnum, flonum or bignum).
.Lf fixp "'g_arg"
.Re
t iff g_arg is a fixnum or bignum.
.Lf floatp "'g_arg"
.Re
t iff g_arg is a flonum.
.Lf evenp "'x_arg"
.Re
t iff x_arg is even.
.Lf oddp "'x_arg"
.Re
t iff x_arg is odd.
.Lf zerop "'g_arg"
.Re
t iff g_arg is a number equal to 0.
.Lf onep "'g_arg"
.Re
t iff g_arg is a number equal to 1.
.Lf plusp "'n_arg"
.Re
t iff n_arg is greater than zero.
.Lf minusp "'g_arg"
.Re
t iff g_arg is a negative number.
.Lf greaterp "['n_arg1 ...]"
.Lx > "'fx_arg1 'fx_arg2"
.Lx >& "'x_arg1 'x_arg2"
.Re
t iff the arguments are in a strictly decreasing order.
.No
In functions
.i greaterp
and
.i >
the function 
.i difference 
is used to compare adjacent values. 
If any of the arguments are non numbers, the error message will come 
from the 
.i difference 
function.
The arguments to 
.i >
must  be fixnums or both flonums.
The arguments to
.i >&
must both be fixnums.
.Lf lessp "['n_arg1 ...]"
.Lx < "'fx_arg1 'fx_arg2"
.Lx <& "'x_arg1 'x_arg2"
.Re
t iff the arguments are in a strictly increasing order.
.No
In functions
.i lessp
and
.i <
the function \fIdifference\fP is used to compare adjacent values. 
If any of the arguments are non numbers, the error message will come 
from the \fIdifference\fP function.
The arguments to 
.i < 
may be either fixnums or flonums but must be the same type.
The arguments to
.i <&
must be fixnums.
.Lf \(eq "'fx_arg1 'fx_arg2"
.Lf \(eq& "'x_arg1 'x_arg2"
.Re
t iff the arguments have the same value.
The arguments to \(eq must be the either both fixnums or both flonums.
The arguments to \(eq& must be fixnums.
.sh 2 trignometric\ functions 
.Lf cos "'fx_angle"
.Re
the (flonum) cosine of fx_angle (which is assumed to be in radians).
.Lf sin "'fx_angle"
.Re
the sine of fx_angle (which is assumed to be in radians).
.Lf acos "'fx_arg"
.Re
the (flonum) arc cosine of fx_arg in the range 0 to \(*p.
.Lf asin "'fx_arg"
.Re
the (flonum) arc sine of fx_arg in the range \(mi\(*p/2 to \(*p/2.
.Lf atan "'fx_arg1 'fx_arg2"
.Re
the (flonum) arc tangent of fx_arg1/fx_arg2 in the range -\(*p to \(*p.
.sh 2 bignum\ functions
.Lf haipart "bx_number x_bits"
.Re
a fixnum which contains
the x_bits high bits of
\fI(abs\ bx_number)\fP if x_bits is positive, otherwise
it returns the \fI(abs\ x_bits)\fP low bits of \fI(abs\ bx_number)\fP.
.Lf haulong "bx_number"
.Re
the number of significant bits in bx_number.
.No
the result is equal to the least integer greater to or equal to the
base two logarithm of
one plus the absolute value of bx_number.
.Lf bignum-leftshift "bx_arg x_amount"
.Re
bx_arg shifted left by x_amount.  If
x_amount is negative, bx_arg will be shifted right by the magnitude of
x_amount.
.No
If bx_arg is shifted right, it will be rounded to the nearest even number.
.Lf sticky-bignum-leftshift "'bx_arg 'x_amount"
.Re
bx_arg shifted left by x_amount.  If
x_amount is negative, bx_arg will be shifted right by the magnitude of
x_amount and rounded.
.No
sticky rounding is done this way: the low order bit is changed from its
value after shifting to 1 if any 1's were shifted off to the right.
.sh 2 bit\ manipulation
.Lf boole "'x_key 'x_v1 'x_v2 ..."
.Re
the result of the bitwise boolean operation as described in the following
table.
.No
If there are more than 3 arguments, then evaluation proceeds left to
right with each partial result becoming the new value of x_v1.
That is, 
.br
\ \ \ \ \ \fI(boole\ 'key\ 'v1\ 'v2\ 'v3)\ \(==\ (boole\ 'key\ (boole\ 'key\ 'v1\ 'v2)\ 'v3)\fP.
.br
In the following table, \(** represents bitwise and, \(pl represents
bitwise or, \o'\(ci\(pl' represents bitwise xor and \(no represents
bitwise negation and is the highest precedence operator.
.ps 8
.TS
center box ;
c s s s s s s s s
c c c c c c c c c.
(boole 'key 'x 'y)

=
key	0	1	2	3	4	5	6	7
result	0	x \(** y	\(no x \(** y	y	x \(** \(no y	x	x \o'\(ci\(pl' y	x \(pl y

_

key	8	9	10	11	12	13	14	15
result	\(no (x \(pl y)	\(no(x \o'\(ci\(pl' y)	\(no x	\(no x \(pl y	\(no y	x \(pl \(no y	\(no x \(pl \(no y	-1
.TE
.ps 10
.Lf lsh "'x_val 'x_amt"
.Re
x_val shifted left by x_amt if x_amt is positive.
If x_amt is negative, then 
.i lsh
returns x_val shifted right by the magnitude if x_amt.
.No
This always returns a fixnum even for those numbers whose magnitude is
so large that they would normally be represented as a bignum.
For more general bit shifters see
.i bignum-leftshift
and
.i sticky-bignum-leftshift.
.Lf rot "'x_val 'x_amt"
.Re
x_val rotated left by x_amt if x_amt is positive. 
If x_amt is negative, then x_val is rotated right by the magnitude of x_amt.
.sh 2 other\ functions
.Lf abs 'n_arg
.Lx absval 'n_arg
.Re
the absolute value of n_arg.
.Lf exp "'fx_arg"
.Re
.i e
raised to the fx_arg power (flonum) .
.Lf expt "'n_base 'n_power"
.Re
n_base raised to the n_power power.
.No
if either of the arguments are flonums, the calculation will be done using
.i log
and 
.i exp .
.Lf fact "'x_arg"
.Re
x_arg factorial. (fixnum or bignum)
.Lf fix "'n_arg"
.Re
a fixnum as close as we can get to n_arg.
.No
\fIfix\fP will round down.
Currently, if n_arg is a flonum larger 
than the size of a fixnum, this will fail.
.Lf float "'n_arg"
.Re
a flonum as close as we can get to n_arg.
.No
if n_arg is a bignum larger than the maximum size of a flonum,
then a floating exception will occur.
.Lf log "'fx_arg"
.Re
the natural logarithm of fx_arg.
.Lf max "'n_arg1 ... "
.Re
the maximum value in the list of arguments.
.Lf min "'n_arg1 ... "
.Re
the minimum value in the list of arguments.
.Lf mod "'i_dividend 'i_divisor"
.Lx remainder "'i_dividend 'i_divisor"
.Re
the remainder when i_dividend is divided by i_divisor.
.No
The sign of the result will have the same sign as i_dividend.
.Lf *mod "'x_dividend 'x_divisor"
.Re
the balanced representation of x_dividend modulo x_divisor.
.No
the range of the balanced representation is abs(x_divisor)/2 to 
abs(x_divisor)/2 - x_divisor + 1.
.Lf random "['x_limit]"
.Re
a fixnum between 0 and x_limit \(mi 1 if x_limit is given.
If x_limit is not given, any fixnum, positive or negative, might be
returned.
.Lf sqrt "'fx_arg"
.Re
the square root of fx_arg.