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EXP(3)		       FreeBSD Library Functions Manual			EXP(3)

     exp, expf,	exp10, exp10f, expm1, expm1f, log, logf, log10,	log10f,	log1p,
     log1pf, pow, powf -- exponential, logarithm, power	functions

     Math Library (libm, -lm)

     #include <math.h>

     exp(double	x);

     expf(float	x);

     expm1(double x);

     expm1f(float x);

     log(double	x);

     logf(float	x);

     log10(double x);

     log10f(float x);

     log1p(double x);

     log1pf(float x);

     pow(double	x, double y);

     powf(float	x, float y);

     The exp() and the expf() functions	compute	the exponential	value of the
     given argument x.

     The expm1() and the expm1f() functions compute the	value exp(x)-1 accu-
     rately even for tiny argument x.

     The log() and the logf() functions	compute	the value of the natural loga-
     rithm of argument x.

     The log10() and the log10f() functions compute the	value of the logarithm
     of	argument x to base 10.

     The log1p() and the log1pf() functions compute the	value of log(1+x)
     accurately	even for tiny argument x.

     The pow() and the powf() functions	compute	the value of x to the exponent

ERROR (due to Roundoff etc.)
     exp(x), log(x), expm1(x) and log1p(x) are accurate	to within an ulp, and
     log10(x) to within	about 2	ulps; an ulp is	one Unit in the	Last Place.
     The error in pow(x, y) is below about 2 ulps when its magnitude is	moder-
     ate, but increases	as pow(x, y) approaches	the over/underflow thresholds
     until almost as many bits could be	lost as	are occupied by	the float-
     ing-point format's	exponent field;	that is	8 bits for VAX D and 11	bits
     for IEEE 754 Double.  No such drastic loss	has been exposed by testing;
     the worst errors observed have been below 20 ulps for VAX D, 300 ulps for
     IEEE 754 Double.  Moderate	values of pow()	are accurate enough that
     pow(integer, integer) is exact until it is	bigger than 2**56 on a VAX,
     2**53 for IEEE 754.

     These functions will return the appropriate computation unless an error
     occurs or an argument is out of range.  The functions exp(), expm1(),
     pow() detect if the computed value	will overflow, set the global variable
     errno to ERANGE and cause a reserved operand fault	on a VAX or Tahoe.
     The functions pow(x, y) checks to see if x	< 0 and	y is not an integer,
     in	the event this is true,	the global variable errno is set to EDOM and
     on	the VAX	and Tahoe generate a reserved operand fault.  On a VAX and
     Tahoe, errno is set to EDOM and the reserved operand is returned by log
     unless x >	0, by log1p() unless x > -1.

     The functions exp(x)-1 and	log(1+x) are called expm1 and logp1 in BASIC
     on	the Hewlett-Packard HP-71B and APPLE Macintosh,	EXP1 and LN1 in	Pas-
     cal, exp1 and log1	in C on	APPLE Macintoshes, where they have been	pro-
     vided to make sure	financial calculations of ((1+x)**n-1)/x, namely
     expm1(n*log1p(x))/x, will be accurate when	x is tiny.  They also provide
     accurate inverse hyperbolic functions.

     The function pow(x, 0) returns x**0 = 1 for all x including x = 0,	Infin-
     ity (not found on a VAX), and NaN (the reserved operand on	a VAX).	 Pre-
     vious implementations of pow may have defined x**0	to be undefined	in
     some or all of these cases.  Here are reasons for returning x**0 =	1

     1.	     Any program that already tests whether x is zero (or infinite or
	     NaN) before computing x**0	cannot care whether 0**0 = 1 or	not.
	     Any program that depends upon 0**0	to be invalid is dubious any-
	     way since that expression's meaning and, if invalid, its conse-
	     quences vary from one computer system to another.

     2.	     Some Algebra texts	(e.g. Sigler's)	define x**0 = 1	for all	x,
	     including x = 0.  This is compatible with the convention that
	     accepts a[0] as the value of polynomial

		   p(x)	= a[0]*x**0 + a[1]*x**1	+ a[2]*x**2 +...+ a[n]*x**n

	     at	x = 0 rather than reject a[0]*0**0 as invalid.

     3.	     Analysts will accept 0**0 = 1 despite that	x**y can approach any-
	     thing or nothing as x and y approach 0 independently.  The	reason
	     for setting 0**0 =	1 anyway is this:

		   If x(z) and y(z) are	any functions analytic (expandable in
		   power series) in z around z = 0, and	if there x(0) =	y(0) =
		   0, then x(z)**y(z) -> 1 as z	-> 0.

     4.	     If	0**0 = 1, then infinity**0 = 1/0**0 = 1	too; and then NaN**0 =
	     1 too because x**0	= 1 for	all finite and infinite	x, i.e., inde-
	     pendently of x.


     A exp(), log() and	pow() functions	appeared in Version 6 AT&T UNIX.  A
     log10() function appeared in Version 7 AT&T UNIX.	The log1p() and
     expm1() functions appeared	in 4.3BSD.

FreeBSD	11.1			 July 31, 1991			  FreeBSD 11.1


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