EXP(3) BSD Programmer's Manual EXP(3)
exp, expf, exp2, exp2f, expm1, expm1f, - exponential functions
#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 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 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.
The exp() functions conform to ANSI X3.159-1989 ("ANSI C"). 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. MirOS BSD #10-current September 13, 2011 1
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