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infnan(3m) [bsd man page]

INFNAN(3M)																INFNAN(3M)

NAME
infnan - signals invalid floating-point operations on a VAX (temporary) SYNOPSIS
#include <math.h> double infnan(iarg) int iarg; DESCRIPTION
At some time in the future, some of the useful properties of the Infinities and NaNs in the IEEE standard 754 for Binary Floating-Point Arithmetic will be simulated in UNIX on the DEC VAX by using its Reserved Operands. Meanwhile, the Invalid, Overflow and Divide-by-Zero exceptions of the IEEE standard are being approximated on a VAX by calls to a procedure infnan in appropriate places in libm. When better exception-handling is implemented in UNIX, only infnan among the codes in libm will have to be changed. And users of libm can design their own infnan now to insulate themselves from future changes. Whenever an elementary function code in libm has to simulate one of the aforementioned IEEE exceptions, it calls infnan(iarg) with an appropriate value of iarg. Then a reserved operand fault stops computation. But infnan could be replaced by a function with the same name that returns some plausible value, assigns an apt value to the global variable errno, and allows computation to resume. Alternatively, the Reserved Operand Fault Handler could be changed to respond by returning that plausible value, etc. instead of aborting. In the table below, the first two columns show various exceptions signaled by the IEEE standard, and the default result it prescribes. The third column shows what value is given to iarg by functions in libm when they invoke infnan(iarg) under analogous circumstances on a VAX. Currently infnan stops computation under all those circumstances. The last two columns offer an alternative; they suggest a setting for errno and a value for a revised infnan to return. And a C program to implement that suggestion follows. IEEE IEEE Signal Default iarg errno infnan __________________________________________________ Invalid NaN EDOM EDOM 0 Overflow +-Infinity ERANGE ERANGEHUGE Div-by-0 +-Infinity +-ERANGE ERANGE or EDOM+-HUGE (HUGE = 1.7e38 ... nearly 2.0**127) ALTERNATIVE infnan: #include <math.h> #include <errno.h> extern int errno ; double infnan(iarg) int iarg ; { switch(iarg) { case ERANGE: errno = ERANGE; return(HUGE); case -ERANGE: errno = EDOM; return(-HUGE); default: errno = EDOM; return(0); } } SEE ALSO
math(3M), intro(2), signal(3). ERANGE and EDOM are defined in <errno.h>. See intro(2) for explanation of EDOM and ERANGE. 4.3 Berkeley Distribution May 27, 1986 INFNAN(3M)

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EXP(3M) 																   EXP(3M)

NAME
exp, expm1, log, log10, log1p, pow - exponential, logarithm, power SYNOPSIS
#include <math.h> double exp(x) double x; double expm1(x) double x; double log(x) double x; double log10(x) double x; double log1p(x) double x; double pow(x,y) double x,y; DESCRIPTION
Exp returns the exponential function of x. Expm1 returns exp(x)-1 accurately even for tiny x. Log returns the natural logarithm of x. Log10 returns the logarithm of x to base 10. Log1p returns log(1+x) accurately even for tiny x. Pow(x,y) returns x**y. 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 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. DIAGNOSTICS
Exp, expm1 and pow return the reserved operand on a VAX when the correct value would overflow, and they set errno to ERANGE. Pow(x,y) returns the reserved operand on a VAX and sets errno to EDOM when x < 0 and y is not an integer. On a VAX, errno is set to EDOM and the reserved operand is returned by log unless x > 0, by log1p unless x > -1. NOTES
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. 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 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 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., indepen- dently of x. SEE ALSO
math(3M), infnan(3M) AUTHOR
Kwok-Choi Ng, W. Kahan 4th Berkeley Distribution May 27, 1986 EXP(3M)

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