4.3BSD/usr/lib/lisp/manual/ch3.r
CHAPTER 3
Arithmetic Functions
This chapter describes FRANZ LISP's functions for doing
arithmetic. Often the same function is known by many names.
For example, _a_d_d is also _p_l_u_s, and _s_u_m. This is caused by
our desire to be compatible with other Lisps. The FRANZ
LISP user should avoid using functions with names such as +
and * unless their arguments are fixnums. The Lisp compiler
takes advantage of these implicit declarations.
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
desired (see the description of the _s_i_g_n_a_l function).
3.1. Simple Arithmetic Functions
(add ['n_arg1 ...])
(plus ['n_arg1 ...])
(sum ['n_arg1 ...])
(+ ['x_arg1 ...])
RETURNS: the sum of the arguments. If no arguments are
given, 0 is returned.
NOTE: 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 pro-
cess of doing the addition a bignum must be con-
verted into a flonum an error message will
result.
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9Arithmetic Functions 3-1
Arithmetic Functions 3-2
(add1 'n_arg)
(1+ 'x_arg)
RETURNS: its argument plus 1.
(diff ['n_arg1 ... ])
(difference ['n_arg1 ... ])
(- ['x_arg1 ... ])
RETURNS: the result of subtracting from n_arg1 all sub-
sequent arguments. If no arguments are given,
0 is returned.
NOTE: See the description of add for details on data
type conversions and restrictions.
(sub1 'n_arg)
(1- 'x_arg)
RETURNS: its argument minus 1.
(minus 'n_arg)
RETURNS: zero minus n_arg.
(product ['n_arg1 ... ])
(times ['n_arg1 ... ])
(* ['x_arg1 ... ])
RETURNS: the product of all of its arguments. It
returns 1 if there are no arguments.
NOTE: See the description of the function _a_d_d for
details and restrictions to the automatic data
type coercion.
(quotient ['n_arg1 ...])
(/ ['x_arg1 ...])
RETURNS: the result of dividing the first argument by
succeeding ones.
NOTE: If there are no arguments, 1 is returned. See
the description of the function _a_d_d for details
and restrictions of data type coercion. A divide
by zero will cause a floating exception interrupt
-- see the description of the _s_i_g_n_a_l function.
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Arithmetic Functions 3-3
(*quo 'i_x 'i_y)
RETURNS: the integer part of i_x / i_y.
(Divide 'i_dividend 'i_divisor)
RETURNS: a list whose car is the quotient and whose
cadr is the remainder of the division of
i_dividend by i_divisor.
NOTE: this is restricted to integer division.
(Emuldiv 'x_fact1 'x_fact2 'x_addn 'x_divisor)
RETURNS: a list of the quotient and remainder of this
operation:
((x_fact1 * x_fact2) + (sign extended) x_addn) / x_divisor.
NOTE: this is useful for creating a bignum arithmetic
package in Lisp.
3.2. predicates
(numberp 'g_arg)
(numbp 'g_arg)
RETURNS: t iff g_arg is a number (fixnum, flonum or
bignum).
(fixp 'g_arg)
RETURNS: t iff g_arg is a fixnum or bignum.
(floatp 'g_arg)
RETURNS: t iff g_arg is a flonum.
(evenp 'x_arg)
RETURNS: t iff x_arg is even.
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Arithmetic Functions 3-4
(oddp 'x_arg)
RETURNS: t iff x_arg is odd.
(zerop 'g_arg)
RETURNS: t iff g_arg is a number equal to 0.
(onep 'g_arg)
RETURNS: t iff g_arg is a number equal to 1.
(plusp 'n_arg)
RETURNS: t iff n_arg is greater than zero.
(minusp 'g_arg)
RETURNS: t iff g_arg is a negative number.
(greaterp ['n_arg1 ...])
(> 'fx_arg1 'fx_arg2)
(>& 'x_arg1 'x_arg2)
RETURNS: t iff the arguments are in a strictly decreas-
ing order.
NOTE: In functions _g_r_e_a_t_e_r_p and > the function _d_i_f_f_e_r_-
_e_n_c_e is used to compare adjacent values. If any
of the arguments are non-numbers, the error mes-
sage will come from the _d_i_f_f_e_r_e_n_c_e function. The
arguments to > must be fixnums or both flonums.
The arguments to >& must both be fixnums.
(lessp ['n_arg1 ...])
(< 'fx_arg1 'fx_arg2)
(<& 'x_arg1 'x_arg2)
RETURNS: t iff the arguments are in a strictly increas-
ing order.
NOTE: In functions _l_e_s_s_p and < the function _d_i_f_f_e_r_e_n_c_e
is used to compare adjacent values. If any of the
arguments are non numbers, the error message will
come from the _d_i_f_f_e_r_e_n_c_e function. The arguments
to < may be either fixnums or flonums but must be
the same type. The arguments to <& must be fix-
nums.
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Arithmetic Functions 3-5
(= 'fx_arg1 'fx_arg2)
(=& 'x_arg1 'x_arg2)
RETURNS: t iff the arguments have the same value. The
arguments to = must be the either both fixnums
or both flonums. The arguments to =& must be
fixnums.
3.3. Trignometric Functions
Some of these funtcions are taken from the host
math library, and we take no further responsibility
for their accuracy.
(cos 'fx_angle)
RETURNS: the (flonum) cosine of fx_angle (which is
assumed to be in radians).
(sin 'fx_angle)
RETURNS: the sine of fx_angle (which is assumed to be
in radians).
(acos 'fx_arg)
RETURNS: the (flonum) arc cosine of fx_arg in the range
0 to J.
(asin 'fx_arg)
RETURNS: the (flonum) arc sine of fx_arg in the range
-J/2 to J/2.
(atan 'fx_arg1 'fx_arg2)
RETURNS: the (flonum) arc tangent of fx_arg1/fx_arg2 in
the range -J to J.
3.4. Bignum/Fixnum Manipulation
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Arithmetic Functions 3-6
(haipart bx_number x_bits)
RETURNS: a fixnum (or bignum) which contains the x_bits
high bits of (_a_b_s _b_x__n_u_m_b_e_r) if x_bits is
positive, otherwise it returns the
(_a_b_s _x__b_i_t_s) low bits of (_a_b_s _b_x__n_u_m_b_e_r).
(haulong bx_number)
RETURNS: the number of significant bits in bx_number.
NOTE: 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.
(bignum-leftshift bx_arg x_amount)
RETURNS: bx_arg shifted left by x_amount. If x_amount
is negative, bx_arg will be shifted right by
the magnitude of x_amount.
NOTE: If bx_arg is shifted right, it will be rounded to
the nearest even number.
(sticky-bignum-leftshift 'bx_arg 'x_amount)
RETURNS: 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.
NOTE: sticky rounding is done this way: after shifting,
the low order bit is changed to 1 if any 1's were
shifted off to the right.
3.5. Bit Manipulation
(boole 'x_key 'x_v1 'x_v2 ...)
RETURNS: the result of the bitwise boolean operation as
described in the following table.
NOTE: If there are more than 3 arguments, then evalua-
tion proceeds left to right with each partial
result becoming the new value of x_v1. That is,
(_b_o_o_l_e '_k_e_y '_v_1 '_v_2 '_v_3) =_ (_b_o_o_l_e '_k_e_y (_b_o_o_l_e '_k_e_y '_v_1 '_v_2) '_v_3).
In the following table, * represents bitwise and,
+ represents bitwise or, O+ represents bitwise xor
and _ represents bitwise negation and is the
highest precedence operator.
Arithmetic Functions 3-7
8____________________________________________________________________________________________
(boole 'key 'x 'y)
8________________________________________________________________________________________________________________________________________________________________________________________
key 0 1 2 3 4 5 6 7
result 0 x * y _ x * y y x * _ y x x O+ y x + y
common
names and bitclear xor or
8____________________________________________________________________________________________
key 8 9 10 11 12 13 14 15
result _ (x + y) _(x O+ y) _ x _ x + y _ y x + _ y _ x + _ y -1
common
names nor equiv implies nand
8____________________________________________________________________________________________
7|8|7|7|7|7|7|7|7|7|7|7|7|7|7|
9 |8|7|7|7|7|7|7|7|7|7|7|7|7|7|
9
(lsh 'x_val 'x_amt)
RETURNS: x_val shifted left by x_amt if x_amt is posi-
tive. If x_amt is negative, then _l_s_h returns
x_val shifted right by the magnitude if x_amt.
NOTE: This always returns a fixnum even for those
numbers whose magnitude is so large that they
would normally be represented as a bignum, i.e.
shifter bits are lost. For more general bit
shifters, see _b_i_g_n_u_m-_l_e_f_t_s_h_i_f_t and _s_t_i_c_k_y-
_b_i_g_n_u_m-_l_e_f_t_s_h_i_f_t.
(rot 'x_val 'x_amt)
RETURNS: x_val rotated left by x_amt if x_amt is posi-
tive. If x_amt is negative, then x_val is
rotated right by the magnitude of x_amt.
3.6. Other Functions
As noted above, some of the following functions
are inherited from the host math library, with all
their virtues and vices.
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Arithmetic Functions 3-8
(abs 'n_arg)
(absval 'n_arg)
RETURNS: the absolute value of n_arg.
(exp 'fx_arg)
RETURNS: _e raised to the fx_arg power (flonum) .
(expt 'n_base 'n_power)
RETURNS: n_base raised to the n_power power.
NOTE: if either of the arguments are flonums, the cal-
culation will be done using _l_o_g and _e_x_p.
(fact 'x_arg)
RETURNS: x_arg factorial. (fixnum or bignum)
(fix 'n_arg)
RETURNS: a fixnum as close as we can get to n_arg.
NOTE: _f_i_x will round down. Currently, if n_arg is a
flonum larger than the size of a fixnum, this
will fail.
(float 'n_arg)
RETURNS: a flonum as close as we can get to n_arg.
NOTE: if n_arg is a bignum larger than the maximum size
of a flonum, then a floating exception will
occur.
(log 'fx_arg)
RETURNS: the natural logarithm of fx_arg.
(max 'n_arg1 ... )
RETURNS: the maximum value in the list of arguments.
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Arithmetic Functions 3-9
(min 'n_arg1 ... )
RETURNS: the minimum value in the list of arguments.
(mod 'i_dividend 'i_divisor)
(remainder 'i_dividend 'i_divisor)
RETURNS: the remainder when i_dividend is divided by
i_divisor.
NOTE: The sign of the result will have the same sign as
i_dividend.
(*mod 'x_dividend 'x_divisor)
RETURNS: the balanced representation of x_dividend
modulo x_divisor.
NOTE: the range of the balanced representation is
abs(x_divisor)/2 to (abs(x_divisor)/2) -
x_divisor + 1.
(random ['x_limit])
RETURNS: a fixnum between 0 and x_limit - 1 if x_limit
is given. If x_limit is not given, any fix-
num, positive or negative, might be returned.
(sqrt 'fx_arg)
RETURNS: the square root of fx_arg.
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