xref: /freebsd/lib/msun/man/exp.3 (revision 1e413cf93298b5b97441a21d9a50fdcd0ee9945e)
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28.\"     from: @(#)exp.3	6.12 (Berkeley) 7/31/91
29.\" $FreeBSD$
30.\"
31.Dd April 5, 2005
32.Dt EXP 3
33.Os
34.Sh NAME
35.Nm exp ,
36.Nm expf ,
37.\" The sorting error is intentional.  exp and expf should be adjacent.
38.Nm exp2 ,
39.Nm exp2f ,
40.Nm expm1 ,
41.Nm expm1f ,
42.Nm log ,
43.Nm logf ,
44.Nm log10 ,
45.Nm log10f ,
46.Nm log1p ,
47.Nm log1pf ,
48.Nm pow ,
49.Nm powf
50.Nd exponential, logarithm, power functions
51.Sh LIBRARY
52.Lb libm
53.Sh SYNOPSIS
54.In math.h
55.Ft double
56.Fn exp "double x"
57.Ft float
58.Fn expf "float x"
59.Ft double
60.Fn exp2 "double x"
61.Ft float
62.Fn exp2f "float x"
63.Ft double
64.Fn expm1 "double x"
65.Ft float
66.Fn expm1f "float x"
67.Ft double
68.Fn log "double x"
69.Ft float
70.Fn logf "float x"
71.Ft double
72.Fn log10 "double x"
73.Ft float
74.Fn log10f "float x"
75.Ft double
76.Fn log1p "double x"
77.Ft float
78.Fn log1pf "float x"
79.Ft double
80.Fn pow "double x" "double y"
81.Ft float
82.Fn powf "float x" "float y"
83.Sh DESCRIPTION
84The
85.Fn exp
86and the
87.Fn expf
88functions compute the base
89.Ms e
90exponential value of the given argument
91.Fa x .
92.Pp
93The
94.Fn exp2
95and the
96.Fn exp2f
97functions compute the base 2 exponential of the given argument
98.Fa x .
99.Pp
100The
101.Fn expm1
102and the
103.Fn expm1f
104functions compute the value exp(x)\-1 accurately even for tiny argument
105.Fa x .
106.Pp
107The
108.Fn log
109and the
110.Fn logf
111functions compute the value of the natural logarithm of argument
112.Fa x .
113.Pp
114The
115.Fn log10
116and the
117.Fn log10f
118functions compute the value of the logarithm of argument
119.Fa x
120to base 10.
121.Pp
122The
123.Fn log1p
124and the
125.Fn log1pf
126functions compute
127the value of log(1+x) accurately even for tiny argument
128.Fa x .
129.Pp
130The
131.Fn pow
132and the
133.Fn powf
134functions compute the value
135of
136.Ar x
137to the exponent
138.Ar y .
139.Sh ERROR (due to Roundoff etc.)
140The values of
141.Fn exp 0 ,
142.Fn expm1 0 ,
143.Fn exp2 integer ,
144and
145.Fn pow integer integer
146are exact provided that they are representable.
147.\" XXX Is this really true for pow()?
148Otherwise the error in these functions is generally below one
149.Em ulp .
150.Sh RETURN VALUES
151These functions will return the appropriate computation unless an error
152occurs or an argument is out of range.
153The functions
154.Fn pow x y
155and
156.Fn powf x y
157raise an invalid exception and return an \*(Na if
158.Fa x
159< 0 and
160.Fa y
161is not an integer.
162An attempt to take the logarithm of \*(Pm0 will result in
163a divide-by-zero exception, and an infinity will be returned.
164An attempt to take the logarithm of a negative number will
165result in an invalid exception, and an \*(Na will be generated.
166.Sh NOTES
167The functions exp(x)\-1 and log(1+x) are called
168expm1 and logp1 in
169.Tn BASIC
170on the Hewlett\-Packard
171.Tn HP Ns \-71B
172and
173.Tn APPLE
174Macintosh,
175.Tn EXP1
176and
177.Tn LN1
178in Pascal, exp1 and log1 in C
179on
180.Tn APPLE
181Macintoshes, where they have been provided to make
182sure financial calculations of ((1+x)**n\-1)/x, namely
183expm1(n\(**log1p(x))/x, will be accurate when x is tiny.
184They also provide accurate inverse hyperbolic functions.
185.Pp
186The function
187.Fn pow x 0
188returns x**0 = 1 for all x including x = 0, \*(If, and \*(Na .
189Previous implementations of pow may
190have defined x**0 to be undefined in some or all of these
191cases.
192Here are reasons for returning x**0 = 1 always:
193.Bl -enum -width indent
194.It
195Any program that already tests whether x is zero (or
196infinite or \*(Na) before computing x**0 cannot care
197whether 0**0 = 1 or not.
198Any program that depends
199upon 0**0 to be invalid is dubious anyway since that
200expression's meaning and, if invalid, its consequences
201vary from one computer system to another.
202.It
203Some Algebra texts (e.g.\& Sigler's) define x**0 = 1 for
204all x, including x = 0.
205This is compatible with the convention that accepts a[0]
206as the value of polynomial
207.Bd -literal -offset indent
208p(x) = a[0]\(**x**0 + a[1]\(**x**1 + a[2]\(**x**2 +...+ a[n]\(**x**n
209.Ed
210.Pp
211at x = 0 rather than reject a[0]\(**0**0 as invalid.
212.It
213Analysts will accept 0**0 = 1 despite that x**y can
214approach anything or nothing as x and y approach 0
215independently.
216The reason for setting 0**0 = 1 anyway is this:
217.Bd -ragged -offset indent
218If x(z) and y(z) are
219.Em any
220functions analytic (expandable
221in power series) in z around z = 0, and if there
222x(0) = y(0) = 0, then x(z)**y(z) \(-> 1 as z \(-> 0.
223.Ed
224.It
225If 0**0 = 1, then
226\*(If**0 = 1/0**0 = 1 too; and
227then \*(Na**0 = 1 too because x**0 = 1 for all finite
228and infinite x, i.e., independently of x.
229.El
230.Sh SEE ALSO
231.Xr fenv 3 ,
232.Xr math 3
233