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1.1 root 1: .\" Copyright (c) 1980 Regents of the University of California.
2: .\" All rights reserved. The Berkeley software License Agreement
3: .\" specifies the terms and conditions for redistribution.
4: .\"
5: .\" @(#)ch3.n 6.2 (Berkeley) 5/14/86
6: .\"
7: ." $Header: ch3.n,v 1.3 83/06/21 13:00:48 sklower Exp $
8: .Lc Arithmetic\ Functions 3
9: .pp
10: This chapter describes
11: .Fr "'s"
12: functions for doing arithmetic.
13: Often the same function is known by many names.
14: For example,
15: .i add
16: is also
17: .i plus ,
18: and
19: .i sum .
20: This is caused by our desire to be compatible with other Lisps.
21: The
22: .Fr
23: user should avoid using functions with names
24: such as \(pl and \(** unless
25: their arguments are fixnums.
26: The Lisp compiler takes advantage of these implicit declarations.
27: .pp
28: An attempt to divide or to generate a floating
29: point result outside of the range of
30: floating point numbers
31: will cause a floating exception signal
32: from the UNIX operating system.
33: The user can catch and process this interrupt if desired (see the
34: description of the
35: .i signal
36: function).
37: .sh 2 Simple\ Arithmetic\ Functions \n(ch 1
38: .Lf add "['n_arg1 ...]"
39: .Lx plus "['n_arg1 ...]"
40: .Lx sum "['n_arg1 ...]"
41: .Lx \(pl "['x_arg1 ...]"
42: .Re
43: the sum of the arguments. If no arguments are given, 0 is returned.
44: .No
45: if the size of the partial sum exceeds the limit of a fixnum, the
46: partial sum will be converted to a bignum.
47: If any of the arguments are flonums, the partial sum will be
48: converted to a flonum when that argument is processed and the
49: result will thus be a flonum.
50: Currently, if in the process of doing the
51: addition a bignum must be converted into
52: a flonum an error message will result.
53: .Lf add1 'n_arg
54: .Lx 1+ 'x_arg
55: .Re
56: its argument plus 1.
57: .Lf diff "['n_arg1 ... ]"
58: .Lx difference "['n_arg1 ... ]"
59: .Lx \(mi "['x_arg1 ... ]"
60: .Re
61: the result of subtracting from n_arg1 all subsequent arguments.
62: If no arguments are given, 0 is returned.
63: .No
64: See the description of add for details on data type conversions and
65: restrictions.
66: .Lf sub1 "'n_arg"
67: .Lx 1\(mi "'x_arg"
68: .Re
69: its argument minus 1.
70: .Lf minus "'n_arg"
71: .Re
72: zero minus n_arg.
73: .Lf product "['n_arg1 ... ]"
74: .Lx times "['n_arg1 ... ]"
75: .Lx \(** "['x_arg1 ... ]"
76: .Re
77: the product of all of its arguments.
78: It returns 1 if there are no arguments.
79: .No
80: See the description of the function \fIadd\fP for details and restrictions to the
81: automatic data type coercion.
82: .Lf quotient "['n_arg1 ...]"
83: .Lx / "['x_arg1 ...]"
84: .Re
85: the result of dividing the first argument by succeeding ones.
86: .No
87: If there are no arguments, 1 is returned.
88: See the description of the function \fIadd\fP for details and restrictions
89: of data type coercion.
90: A divide by zero will cause a floating exception interrupt -- see
91: the description of the
92: .i signal
93: function.
94: .Lf *quo "'i_x 'i_y"
95: .Re
96: the integer part of i_x / i_y.
97: .Lf Divide "'i_dividend 'i_divisor"
98: .Re
99: a list whose car is the quotient and whose cadr is the remainder of the
100: division of i_dividend by i_divisor.
101: .No
102: this is restricted to integer division.
103: .Lf Emuldiv "'x_fact1 'x_fact2 'x_addn 'x_divisor"
104: .Re
105: a list of the quotient and remainder of this operation:
106: ((x_fact1\ *\ x_fact2)\ +\ (sign\ extended)\ x_addn)\ /\ x_divisor.
107: .No
108: this is useful for creating a bignum arithmetic package in Lisp.
109: .sh 2 predicates
110: .Lf numberp "'g_arg"
111: .Lf numbp "'g_arg"
112: .Re
113: t iff g_arg is a number (fixnum, flonum or bignum).
114: .Lf fixp "'g_arg"
115: .Re
116: t iff g_arg is a fixnum or bignum.
117: .Lf floatp "'g_arg"
118: .Re
119: t iff g_arg is a flonum.
120: .Lf evenp "'x_arg"
121: .Re
122: t iff x_arg is even.
123: .Lf oddp "'x_arg"
124: .Re
125: t iff x_arg is odd.
126: .Lf zerop "'g_arg"
127: .Re
128: t iff g_arg is a number equal to 0.
129: .Lf onep "'g_arg"
130: .Re
131: t iff g_arg is a number equal to 1.
132: .Lf plusp "'n_arg"
133: .Re
134: t iff n_arg is greater than zero.
135: .Lf minusp "'g_arg"
136: .Re
137: t iff g_arg is a negative number.
138: .Lf greaterp "['n_arg1 ...]"
139: .Lx > "'fx_arg1 'fx_arg2"
140: .Lx >& "'x_arg1 'x_arg2"
141: .Re
142: t iff the arguments are in a strictly decreasing order.
143: .No
144: In functions
145: .i greaterp
146: and
147: .i >
148: the function
149: .i difference
150: is used to compare adjacent values.
151: If any of the arguments are non-numbers, the error message will come
152: from the
153: .i difference
154: function.
155: The arguments to
156: .i >
157: must be fixnums or both flonums.
158: The arguments to
159: .i >&
160: must both be fixnums.
161: .Lf lessp "['n_arg1 ...]"
162: .Lx < "'fx_arg1 'fx_arg2"
163: .Lx <& "'x_arg1 'x_arg2"
164: .Re
165: t iff the arguments are in a strictly increasing order.
166: .No
167: In functions
168: .i lessp
169: and
170: .i <
171: the function \fIdifference\fP is used to compare adjacent values.
172: If any of the arguments are non numbers, the error message will come
173: from the \fIdifference\fP function.
174: The arguments to
175: .i <
176: may be either fixnums or flonums but must be the same type.
177: The arguments to
178: .i <&
179: must be fixnums.
180: .Lf \(eq "'fx_arg1 'fx_arg2"
181: .Lf \(eq& "'x_arg1 'x_arg2"
182: .Re
183: t iff the arguments have the same value.
184: The arguments to \(eq must be the either both fixnums or both flonums.
185: The arguments to \(eq& must be fixnums.
186: .sh 2 Trignometric\ Functions
187: .pp
188: Some of these funtcions are taken from the host math library, and
189: we take no further responsibility for their accuracy.
190: .Lf cos "'fx_angle"
191: .Re
192: the (flonum) cosine of fx_angle (which is assumed to be in radians).
193: .Lf sin "'fx_angle"
194: .Re
195: the sine of fx_angle (which is assumed to be in radians).
196: .Lf acos "'fx_arg"
197: .Re
198: the (flonum) arc cosine of fx_arg in the range 0 to \(*p.
199: .Lf asin "'fx_arg"
200: .Re
201: the (flonum) arc sine of fx_arg in the range \(mi\(*p/2 to \(*p/2.
202: .Lf atan "'fx_arg1 'fx_arg2"
203: .Re
204: the (flonum) arc tangent of fx_arg1/fx_arg2 in the range -\(*p to \(*p.
205: .sh 2 Bignum/Fixnum\ Manipulation
206: .Lf haipart "bx_number x_bits"
207: .Re
208: a fixnum (or bignum) which contains
209: the x_bits high bits of
210: \fI(abs\ bx_number)\fP if x_bits is positive, otherwise
211: it returns the \fI(abs\ x_bits)\fP low bits of \fI(abs\ bx_number)\fP.
212: .Lf haulong "bx_number"
213: .Re
214: the number of significant bits in bx_number.
215: .No
216: the result is equal to the least integer greater to or equal to the
217: base two logarithm of
218: one plus the absolute value of bx_number.
219: .Lf bignum-leftshift "bx_arg x_amount"
220: .Re
221: bx_arg shifted left by x_amount. If
222: x_amount is negative, bx_arg will be shifted right by the magnitude of
223: x_amount.
224: .No
225: If bx_arg is shifted right, it will be rounded to the nearest even number.
226: .Lf sticky-bignum-leftshift "'bx_arg 'x_amount"
227: .Re
228: bx_arg shifted left by x_amount. If
229: x_amount is negative, bx_arg will be shifted right by the magnitude of
230: x_amount and rounded.
231: .No
232: sticky rounding is done this way: after shifting,
233: the low order bit is changed to 1
234: if any 1's were shifted off to the right.
235: .sh 2 Bit\ Manipulation
236: .Lf boole "'x_key 'x_v1 'x_v2 ..."
237: .Re
238: the result of the bitwise boolean operation as described in the following
239: table.
240: .No
241: If there are more than 3 arguments, then evaluation proceeds left to
242: right with each partial result becoming the new value of x_v1.
243: That is,
244: .br
245: \ \ \ \ \ \fI(boole\ 'key\ 'v1\ 'v2\ 'v3)\ \(==\ (boole\ 'key\ (boole\ 'key\ 'v1\ 'v2)\ 'v3)\fP.
246: .br
247: In the following table, \(** represents bitwise and, \(pl represents
248: bitwise or, \o'\(ci\(pl' represents bitwise xor and \(no represents
249: bitwise negation and is the highest precedence operator.
250: .ps 8
251: .(b
252: .TS
253: center box ;
254: c s s s s s s s s
255: c c c c c c c c c.
256: (boole 'key 'x 'y)
257:
258: =
259: key 0 1 2 3 4 5 6 7
260: result 0 x \(** y \(no x \(** y y x \(** \(no y x x \o'\(ci\(pl' y x \(pl y
261:
262: common
263: names and bitclear xor or
264:
265: _
266:
267: key 8 9 10 11 12 13 14 15
268: 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
269: common
270: names nor equiv implies nand
271: .TE
272: .)b
273: .ps 10
274: .pp
275: .Lf lsh "'x_val 'x_amt"
276: .Re
277: x_val shifted left by x_amt if x_amt is positive.
278: If x_amt is negative, then
279: .i lsh
280: returns x_val shifted right by the magnitude if x_amt.
281: .No
282: This always returns a fixnum even for those numbers whose magnitude is
283: so large that they would normally be represented as a bignum,
284: i.e. shifter bits are lost.
285: For more general bit shifters, see
286: .i bignum-leftshift
287: and
288: .i sticky-bignum-leftshift.
289: .Lf rot "'x_val 'x_amt"
290: .Re
291: x_val rotated left by x_amt if x_amt is positive.
292: If x_amt is negative, then x_val is rotated right by the magnitude of x_amt.
293: .sh 2 Other\ Functions
294: .pp
295: As noted above, some of the following functions are inherited from the
296: host math library, with all their virtues and vices.
297: .Lf abs 'n_arg
298: .Lx absval 'n_arg
299: .Re
300: the absolute value of n_arg.
301: .Lf exp "'fx_arg"
302: .Re
303: .i e
304: raised to the fx_arg power (flonum) .
305: .Lf expt "'n_base 'n_power"
306: .Re
307: n_base raised to the n_power power.
308: .No
309: if either of the arguments are flonums, the calculation will be done using
310: .i log
311: and
312: .i exp .
313: .Lf fact "'x_arg"
314: .Re
315: x_arg factorial. (fixnum or bignum)
316: .Lf fix "'n_arg"
317: .Re
318: a fixnum as close as we can get to n_arg.
319: .No
320: \fIfix\fP will round down.
321: Currently, if n_arg is a flonum larger
322: than the size of a fixnum, this will fail.
323: .Lf float "'n_arg"
324: .Re
325: a flonum as close as we can get to n_arg.
326: .No
327: if n_arg is a bignum larger than the maximum size of a flonum,
328: then a floating exception will occur.
329: .Lf log "'fx_arg"
330: .Re
331: the natural logarithm of fx_arg.
332: .Lf max "'n_arg1 ... "
333: .Re
334: the maximum value in the list of arguments.
335: .Lf min "'n_arg1 ... "
336: .Re
337: the minimum value in the list of arguments.
338: .Lf mod "'i_dividend 'i_divisor"
339: .Lx remainder "'i_dividend 'i_divisor"
340: .Re
341: the remainder when i_dividend is divided by i_divisor.
342: .No
343: The sign of the result will have the same sign as i_dividend.
344: .Lf *mod "'x_dividend 'x_divisor"
345: .Re
346: the balanced representation of x_dividend modulo x_divisor.
347: .No
348: the range of the balanced representation is abs(x_divisor)/2 to
349: (abs(x_divisor)/2) \(mi x_divisor + 1.
350: .Lf random "['x_limit]"
351: .Re
352: a fixnum between 0 and x_limit \(mi 1 if x_limit is given.
353: If x_limit is not given, any fixnum, positive or negative, might be
354: returned.
355: .Lf sqrt "'fx_arg"
356: .Re
357: the square root of fx_arg.
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