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32 .\" from: @(#)exp.3 6.12 (Berkeley) 7/31/91
33 .\" $FreeBSD: src/lib/msun/man/exp.3,v 1.9.2.4 2001/12/17 10:08:36 ru Exp $
53 .Nd exponential, logarithm, power functions
79 .Fn pow "double x" "double y"
81 .Fn powf "float x" "float y"
87 functions compute the exponential value of the given argument
94 functions compute the value exp(x)\-1 accurately even for tiny argument
101 functions compute the value of the natural logarithm of argument
108 functions compute the value of the logarithm of argument
117 the value of log(1+x) accurately even for tiny argument
124 functions compute the value
129 .Sh ERROR (due to Roundoff etc.)
135 are accurate to within
154 magnitude is moderate, but increases as
157 the over/underflow thresholds until almost as many bits could be
158 lost as are occupied by the floating\-point format's exponent
159 field; that is 8 bits for
161 and 11 bits for IEEE 754 Double.
162 No such drastic loss has been exposed by testing; the worst
163 errors observed have been below 20
174 are accurate enough that
175 .Fn pow integer integer
176 is exact until it is bigger than 2**56 on a
182 These functions will return the appropriate computation unless an error
183 occurs or an argument is out of range.
188 detect if the computed value will overflow,
189 set the global variable
193 and cause a reserved operand fault on a
203 is not an integer, in the event this is true,
212 generate a reserved operand fault.
220 and the reserved operand is returned
229 The functions exp(x)\-1 and log(1+x) are called
232 on the Hewlett\-Packard
240 in Pascal, exp1 and log1 in C
243 Macintoshes, where they have been provided to make
244 sure financial calculations of ((1+x)**n\-1)/x, namely
245 expm1(n\(**log1p(x))/x, will be accurate when x is tiny.
246 They also provide accurate inverse hyperbolic functions.
250 returns x**0 = 1 for all x including x = 0,
262 Previous implementations of pow may
263 have defined x**0 to be undefined in some or all of these
264 cases. Here are reasons for returning x**0 = 1 always:
265 .Bl -enum -width indent
267 Any program that already tests whether x is zero (or
268 infinite or \*(Na) before computing x**0 cannot care
269 whether 0**0 = 1 or not.
270 Any program that depends
271 upon 0**0 to be invalid is dubious anyway since that
272 expression's meaning and, if invalid, its consequences
273 vary from one computer system to another.
275 Some Algebra texts (e.g. Sigler's) define x**0 = 1 for
276 all x, including x = 0.
277 This is compatible with the convention that accepts a[0]
278 as the value of polynomial
279 .Bd -literal -offset indent
280 p(x) = a[0]\(**x**0 + a[1]\(**x**1 + a[2]\(**x**2 +...+ a[n]\(**x**n
283 at x = 0 rather than reject a[0]\(**0**0 as invalid.
285 Analysts will accept 0**0 = 1 despite that x**y can
286 approach anything or nothing as x and y approach 0
288 The reason for setting 0**0 = 1 anyway is this:
289 .Bd -ragged -offset indent
292 functions analytic (expandable
293 in power series) in z around z = 0, and if there
294 x(0) = y(0) = 0, then x(z)**y(z) \(-> 1 as z \(-> 0.
299 infinity**0 = 1/0**0 = 1 too; and
301 \(if**0 = 1/0**0 = 1 too; and
302 then \*(Na**0 = 1 too because x**0 = 1 for all finite
303 and infinite x, i.e., independently of x.
325 functions appeared in