EXP(3) OpenBSD Programmer's Manual EXP(3)NAME
exp, expf, exp2, exp2f, exp2l, expm1, expm1f, log, logf, log2, log2f,
log10, log10f, log1p, log1pf, pow, powf - exponential, logarithm, power
exp2l(long double x);
pow(double x, double y);
powf(float x, float y);
The exp() function computes the base e exponential value of the given
argument x. The expf() function is a single precision version of exp().
The exp2() function computes the base 2 exponential of the given argument
x. The exp2f() function is a single precision version of exp2(). The
exp2l() function is an extended precision version of exp2().
The expm1() function computes the value exp(x)-1 accurately even for tiny
argument x. The expm1f() function is a single precision version of
The log() function computes the value of the natural logarithm of
argument x. The logf() function is a single precision version of log().
The log2() function computes the value of the logarithm of argument x to
base 2. The log2f() function is a single precision version of log2().
The log10() function computes the value of the logarithm of argument x to
base 10. The log10f() function is a single precision version of log10().
The log1p() function computes the value of log(1+x) accurately even for
tiny argument x. The log1pf() function is a single precision version of
The pow() function computes the value of x to the exponent y. The powf()
function is a single precision version of pow().
These functions will return the appropriate computation unless an error
occurs or an argument is out of range. The functions exp(), expm1() and
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 function 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.
ERRORS (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
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.
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 as the value of polynomial
p(x) = a*x**0 + a*x**1 + a*x**2 +...+ a[n]*x**n
at x = 0 rather than reject a*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 ALSOinfnan(3), math(3)HISTORY
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.
OpenBSD 4.9 October 27, 2009 OpenBSD 4.9