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NAME    [Toc]    [Back]

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

LIBRARY    [Toc]    [Back]

     Math Library (libm, -lm)

SYNOPSIS    [Toc]    [Back]

     #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);

DESCRIPTION    [Toc]    [Back]

     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 accurately
 even for tiny argument x.

     The log() and the logf() functions compute the value of the natural logarithm
 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.)    [Toc]    [Back]

     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 moderate,
 but increases as pow(x, y) approaches the over/underflow thresholds
     until almost as many bits could be lost as are occupied by the floating-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.

RETURN VALUES    [Toc]    [Back]

     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.

NOTES    [Toc]    [Back]

     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 Pascal,
 exp1 and log1 in C on APPLE Macintoshes, where they have been provided
 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, Infinity
 (not found on a VAX), and NaN (the reserved operand on a VAX).  Previous
 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 anyway
 since that expression's meaning and, if invalid, its consequences
 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 anything
 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., independently
 of x.

SEE ALSO    [Toc]    [Back]


HISTORY    [Toc]    [Back]

     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 5.2.1			 July 31, 1991			 FreeBSD 5.2.1
[ Back ]
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