1.\" Copyright (c) 1985, 1991 Regents of the University of California.
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32.\" from: @(#)exp.3 6.12 (Berkeley) 7/31/91
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33.\" $FreeBSD: head/lib/msun/man/exp.3 84402 2001-10-03 06:25:55Z bde $
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| 33.\" $FreeBSD: head/lib/msun/man/exp.3 84881 2001-10-13 12:23:23Z bde $ |
34.\"
35.Dd July 31, 1991
36.Dt EXP 3
37.Os
38.Sh NAME
39.Nm exp ,
40.Nm expf ,
41.Nm exp10 ,
42.Nm exp10f ,
43.Nm expm1 ,
44.Nm expm1f ,
45.Nm log ,
46.Nm logf ,
47.Nm log10 ,
48.Nm log10f ,
49.Nm log1p ,
50.Nm log1pf ,
51.Nm pow ,
52.Nm powf
53.Nd exponential, logarithm, power functions
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54.Sh LIBRARY
55.Lb libm |
56.Sh SYNOPSIS
57.In math.h
58.Ft double
59.Fn exp "double x"
60.Ft float
61.Fn expf "float x"
62.Ft double
63.Fn expm1 "double x"
64.Ft float
65.Fn expm1f "float x"
66.Ft double
67.Fn log "double x"
68.Ft float
69.Fn logf "float x"
70.Ft double
71.Fn log10 "double x"
72.Ft float
73.Fn log10f "float x"
74.Ft double
75.Fn log1p "double x"
76.Ft float
77.Fn log1pf "float x"
78.Ft double
79.Fn pow "double x" "double y"
80.Ft float
81.Fn powf "float x" "float y"
82.Sh DESCRIPTION
83The
84.Fn exp
85and the
86.Fn expf
87functions compute the exponential value of the given argument
88.Fa x .
89.Pp
90The
91.Fn expm1
92and the
93.Fn expm1f
94functions compute the value exp(x)\-1 accurately even for tiny argument
95.Fa x .
96.Pp
97The
98.Fn log
99and the
100.Fn logf
101functions compute the value of the natural logarithm of argument
102.Fa x .
103.Pp
104The
105.Fn log10
106and the
107.Fn log10f
108functions compute the value of the logarithm of argument
109.Fa x
110to base 10.
111.Pp
112The
113.Fn log1p
114and the
115.Fn log1pf
116functions compute
117the value of log(1+x) accurately even for tiny argument
118.Fa x .
119.Pp
120The
121.Fn pow
122and the
123.Fn powf
124functions compute the value
125of
126.Ar x
127to the exponent
128.Ar y .
129.Sh ERROR (due to Roundoff etc.)
130.Fn exp x ,
131.Fn log x ,
132.Fn expm1 x
133and
134.Fn log1p x
135are accurate to within
136an
137.Em ulp ,
138and
139.Fn log10 x
140to within about 2
141.Em ulps ;
142an
143.Em ulp
144is one
145.Em Unit
146in the
147.Em Last
148.Em Place .
149The error in
150.Fn pow x y
151is below about 2
152.Em ulps
153when its
154magnitude is moderate, but increases as
155.Fn pow x y
156approaches
157the over/underflow thresholds until almost as many bits could be
158lost as are occupied by the floating\-point format's exponent
159field; that is 8 bits for
160.Tn "VAX D"
161and 11 bits for IEEE 754 Double.
162No such drastic loss has been exposed by testing; the worst
163errors observed have been below 20
164.Em ulps
165for
166.Tn "VAX D" ,
167300
168.Em ulps
169for
170.Tn IEEE
171754 Double.
172Moderate values of
173.Fn pow
174are accurate enough that
175.Fn pow integer integer
176is exact until it is bigger than 2**56 on a
177.Tn VAX ,
1782**53 for
179.Tn IEEE
180754.
181.Sh RETURN VALUES
182These functions will return the appropriate computation unless an error
183occurs or an argument is out of range.
184The functions
185.Fn exp ,
186.Fn expm1 ,
187.Fn pow
188detect if the computed value will overflow,
189set the global variable
190.Va errno
191to
192.Er ERANGE
193and cause a reserved operand fault on a
194.Tn VAX
195or
196.Tn Tahoe .
197The functions
198.Fn pow x y
199checks to see if
200.Fa x
201< 0 and
202.Fa y
203is not an integer, in the event this is true,
204the global variable
205.Va errno
206is set to
207.Er EDOM
208and on the
209.Tn VAX
210and
211.Tn Tahoe
212generate a reserved operand fault.
213On a
214.Tn VAX
215and
216.Tn Tahoe ,
217.Va errno
218is set to
219.Er EDOM
220and the reserved operand is returned
221by log unless
222.Fa x
223> 0, by
224.Fn log1p
225unless
226.Fa x
227> \-1.
228.Sh NOTES
229The functions exp(x)\-1 and log(1+x) are called
230expm1 and logp1 in
231.Tn BASIC
232on the Hewlett\-Packard
233.Tn HP Ns \-71B
234and
235.Tn APPLE
236Macintosh,
237.Tn EXP1
238and
239.Tn LN1
240in Pascal, exp1 and log1 in C
241on
242.Tn APPLE
243Macintoshes, where they have been provided to make
244sure financial calculations of ((1+x)**n\-1)/x, namely
245expm1(n\(**log1p(x))/x, will be accurate when x is tiny.
246They also provide accurate inverse hyperbolic functions.
247.Pp
248The function
249.Fn pow x 0
250returns x**0 = 1 for all x including x = 0,
251.if n \
252Infinity
253.if t \
254\(if
255(not found on a
256.Tn VAX ) ,
257and
258.Em NaN
259(the reserved
260operand on a
261.Tn VAX ) .
262Previous implementations of pow may
263have defined x**0 to be undefined in some or all of these
264cases. Here are reasons for returning x**0 = 1 always:
265.Bl -enum -width indent
266.It
267Any program that already tests whether x is zero (or
268infinite or \*(Na) before computing x**0 cannot care
269whether 0**0 = 1 or not.
270Any program that depends
271upon 0**0 to be invalid is dubious anyway since that
272expression's meaning and, if invalid, its consequences
273vary from one computer system to another.
274.It
275Some Algebra texts (e.g. Sigler's) define x**0 = 1 for
276all x, including x = 0.
277This is compatible with the convention that accepts a[0]
278as the value of polynomial
279.Bd -literal -offset indent
280p(x) = a[0]\(**x**0 + a[1]\(**x**1 + a[2]\(**x**2 +...+ a[n]\(**x**n
281.Ed
282.Pp
283at x = 0 rather than reject a[0]\(**0**0 as invalid.
284.It
285Analysts will accept 0**0 = 1 despite that x**y can
286approach anything or nothing as x and y approach 0
287independently.
288The reason for setting 0**0 = 1 anyway is this:
289.Bd -ragged -offset indent
290If x(z) and y(z) are
291.Em any
292functions analytic (expandable
293in power series) in z around z = 0, and if there
294x(0) = y(0) = 0, then x(z)**y(z) \(-> 1 as z \(-> 0.
295.Ed
296.It
297If 0**0 = 1, then
298.if n \
299infinity**0 = 1/0**0 = 1 too; and
300.if t \
301\(if**0 = 1/0**0 = 1 too; and
302then \*(Na**0 = 1 too because x**0 = 1 for all finite
303and infinite x, i.e., independently of x.
304.El
305.Sh SEE ALSO
306.Xr math 3
307.Sh HISTORY
308A
309.Fn exp ,
310.Fn log
311and
312.Fn pow
313functions
314appeared in
315.At v6 .
316A
317.Fn log10
318function
319appeared in
320.At v7 .
321The
322.Fn log1p
323and
324.Fn expm1
325functions appeared in
326.Bx 4.3 .
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