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28 .\" from: @(#)exp.3 6.12 (Berkeley) 7/31/91
37 .\" The sorting error is intentional. exp and expf should be adjacent.
50 .Nd exponential, logarithm, power functions
80 .Fn pow "double x" "double y"
82 .Fn powf "float x" "float y"
88 functions compute the base
90 exponential value of the given argument
97 functions compute the base 2 exponential of the given argument
104 functions compute the value exp(x)\-1 accurately even for tiny argument
111 functions compute the value of the natural logarithm of argument
118 functions compute the value of the logarithm of argument
127 the value of log(1+x) accurately even for tiny argument
134 functions compute the value
139 .Sh ERROR (due to Roundoff etc.)
145 .Fn pow integer integer
146 are exact provided that they are representable.
147 .\" XXX Is this really true for pow()?
148 Otherwise the error in these functions is generally below one
151 These functions will return the appropriate computation unless an error
152 occurs or an argument is out of range.
157 raise an invalid exception and return an \*(Na if
162 An attempt to take the logarithm of \*(Pm0 will result in
163 a divide-by-zero exception, and an infinity will be returned.
164 An attempt to take the logarithm of a negative number will
165 result in an invalid exception, and an \*(Na will be generated.
167 The functions exp(x)\-1 and log(1+x) are called
170 on the Hewlett\-Packard
178 in Pascal, exp1 and log1 in C
181 Macintoshes, where they have been provided to make
182 sure financial calculations of ((1+x)**n\-1)/x, namely
183 expm1(n\(**log1p(x))/x, will be accurate when x is tiny.
184 They also provide accurate inverse hyperbolic functions.
188 returns x**0 = 1 for all x including x = 0, \*(If, and \*(Na .
189 Previous implementations of pow may
190 have defined x**0 to be undefined in some or all of these
192 Here are reasons for returning x**0 = 1 always:
193 .Bl -enum -width indent
195 Any program that already tests whether x is zero (or
196 infinite or \*(Na) before computing x**0 cannot care
197 whether 0**0 = 1 or not.
198 Any program that depends
199 upon 0**0 to be invalid is dubious anyway since that
200 expression's meaning and, if invalid, its consequences
201 vary from one computer system to another.
203 Some Algebra texts (e.g.\& Sigler's) define x**0 = 1 for
204 all x, including x = 0.
205 This is compatible with the convention that accepts a[0]
206 as the value of polynomial
207 .Bd -literal -offset indent
208 p(x) = a[0]\(**x**0 + a[1]\(**x**1 + a[2]\(**x**2 +...+ a[n]\(**x**n
211 at x = 0 rather than reject a[0]\(**0**0 as invalid.
213 Analysts will accept 0**0 = 1 despite that x**y can
214 approach anything or nothing as x and y approach 0
216 The reason for setting 0**0 = 1 anyway is this:
217 .Bd -ragged -offset indent
220 functions analytic (expandable
221 in power series) in z around z = 0, and if there
222 x(0) = y(0) = 0, then x(z)**y(z) \(-> 1 as z \(-> 0.
226 \*(If**0 = 1/0**0 = 1 too; and
227 then \*(Na**0 = 1 too because x**0 = 1 for all finite
228 and infinite x, i.e., independently of x.