EXP(3) BSD Programmer's Manual EXP(3)
exp, expf, exp2, exp2f, expm1, expm1f, - exponential functions
libm
#include <math.h>
double
exp(double x);
float
expf(float x);
double
exp2(double x);
float
exp2f(float x);
double
expm1(double x);
float
expm1f(float x);
The exp() and the expf() functions compute the base e exponential value
of the given argument x.
The exp2() and exp2f() functions compute the base 2 exponential of the
given argument x.
The expm1() and the expm1f() functions computes the value exp(x)-1 accu-
rately even for tiny argument x.
These functions will return the appropriate computation unless an error
occurs or an argument is out of range. The functions exp() and expm1()
detect if the computed value will overflow, set the global variable errno
to ERANGE and cause a reserved operand fault on a VAX. 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
generate a reserved operand fault. On a VAX, errno is set to EDOM and the
reserved operand is returned by log unless x > 0, by log1p() unless x >
-1.
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 thres-
holds 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 test-
ing; 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 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). Previ-
ous 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 always:
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, in-
cluding x = 0. This is compatible with the convention that ac-
cepts 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., in-
dependently of x.
math(3)
The exp() functions conform to ANSI X3.159-1989 ("ANSI C89"). The exp2(),
exp2f(), expf(), expm1(), and expm1f() functions conform to ISO/IEC
9899:1999 ("ISO C99").
The exp() functions appeared in Version 6 AT&T UNIX. The expm1() function
appeared in 4.3BSD.
MirBSD #10-current February 9, 2014 1