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Test Your Knowledge in Computers #1008
Difficulty: Medium
A power series will converge for some values of the variable x and may diverge for others.
True or False?
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scalb(3m) [bsd man page]

IEEE(3M)																  IEEE(3M)

NAME
copysign, drem, finite, logb, scalb - copysign, remainder, exponent manipulations SYNOPSIS
#include <math.h> double copysign(x,y) double x,y; double drem(x,y) double x,y; int finite(x) double x; double logb(x) double x; double scalb(x,n) double x; int n; DESCRIPTION
These functions are required for, or recommended by the IEEE standard 754 for floating-point arithmetic. Copysign(x,y) returns x with its sign changed to y's. Drem(x,y) returns the remainder r := x - n*y where n is the integer nearest the exact value of x/y; moreover if |n-x/y|=1/2 then n is even. Consequently the remainder is computed exactly and |r| <= |y|/2. But drem(x,0) is exceptional; see below under DIAGNOSTICS. Finite(x) = 1 just when -infinity < x < +infinity, = 0 otherwise (when |x| = infinity or x is NaN or x is the VAX's reserved operand.) Logb(x) returns x's exponent n, a signed integer converted to double-precision floating-point and so chosen that 1 <= |x|/2**n < 2 unless x = 0 or (only on machines that conform to IEEE 754) |x| = infinity or x lies between 0 and the Underflow Threshold; see below under "BUGS". Scalb(x,n) = x*(2**n) computed, for integer n, without first computing 2**n. DIAGNOSTICS
IEEE 754 defines drem(x,0) and drem(infinity,y) to be invalid operations that produce a NaN. On a VAX, drem(x,0) returns the reserved op- erand. No infinity exists on a VAX. IEEE 754 defines logb(+-infinity) = +infinity and logb(0) = -infinity, requires the latter to signal Division-by-Zero. But on a VAX, logb(0) = 1.0 - 2.0**31 = -2,147,483,647.0. And if the correct value of scalb(x,n) would overflow on a VAX, it returns the reserved oper- and and sets errno to ERANGE. SEE ALSO
floor(3M), math(3M), infnan(3M) AUTHOR
Kwok-Choi Ng BUGS
Should drem(x,0) and logb(0) on a VAX signal invalidity by setting errno = EDOM? Should logb(0) return -1.7e38? IEEE 754 currently specifies that logb(denormalized no.) = logb(tiniest normalized no. > 0) but the consensus has changed to the specifica- tion in the new proposed IEEE standard p854, namely that logb(x) satisfy 1 <= scalb(|x|,-logb(x)) < Radix ... = 2 for IEEE 754 for every x except 0, infinity and NaN. Almost every program that assumes 754's specification will work correctly if logb follows 854's specification instead. IEEE 754 requires copysign(x,NaN) = +-x but says nothing else about the sign of a NaN. A NaN (Not a Number) is similar in spirit to the VAX's reserved operand, but very different in important details. Since the sign bit of a reserved operand makes it look negative, copysign(x,reserved operand) = -x; should this return the reserved operand instead? 4.3 Berkeley Distribution May 12, 1986 IEEE(3M)

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IEEE(3M)																  IEEE(3M)

NAME
copysign, drem, finite, logb, scalb - copysign, remainder, exponent manipulations SYNOPSIS
#include <math.h> double copysign(x,y) double x,y; double drem(x,y) double x,y; int finite(x) double x; double logb(x) double x; double scalb(x,n) double x; int n; DESCRIPTION
These functions are required for, or recommended by the IEEE standard 754 for floating-point arithmetic. Copysign(x,y) returns x with its sign changed to y's. Drem(x,y) returns the remainder r := x - n*y where n is the integer nearest the exact value of x/y; moreover if |n-x/y|=1/2 then n is even. Consequently the remainder is computed exactly and |r| <= |y|/2. But drem(x,0) is exceptional; see below under DIAGNOSTICS. Finite(x) = 1 just when -infinity < x < +infinity, = 0 otherwise (when |x| = infinity or x is NaN or x is the VAX's reserved operand.) Logb(x) returns x's exponent n, a signed integer converted to double-precision floating-point and so chosen that 1 <= |x|/2**n < 2 unless x = 0 or (only on machines that conform to IEEE 754) |x| = infinity or x lies between 0 and the Underflow Threshold; see below under "BUGS". Scalb(x,n) = x*(2**n) computed, for integer n, without first computing 2**n. DIAGNOSTICS
IEEE 754 defines drem(x,0) and drem(infinity,y) to be invalid operations that produce a NaN. On a VAX, drem(x,0) returns the reserved op- erand. No infinity exists on a VAX. IEEE 754 defines logb(+-infinity) = +infinity and logb(0) = -infinity, requires the latter to signal Division-by-Zero. But on a VAX, logb(0) = 1.0 - 2.0**31 = -2,147,483,647.0. And if the correct value of scalb(x,n) would overflow on a VAX, it returns the reserved oper- and and sets errno to ERANGE. SEE ALSO
floor(3M), math(3M), infnan(3M) AUTHOR
Kwok-Choi Ng BUGS
Should drem(x,0) and logb(0) on a VAX signal invalidity by setting errno = EDOM? Should logb(0) return -1.7e38? IEEE 754 currently specifies that logb(denormalized no.) = logb(tiniest normalized no. > 0) but the consensus has changed to the specifica- tion in the new proposed IEEE standard p854, namely that logb(x) satisfy 1 <= scalb(|x|,-logb(x)) < Radix ... = 2 for IEEE 754 for every x except 0, infinity and NaN. Almost every program that assumes 754's specification will work correctly if logb follows 854's specification instead. IEEE 754 requires copysign(x,NaN) = +-x but says nothing else about the sign of a NaN. A NaN (Not a Number) is similar in spirit to the VAX's reserved operand, but very different in important details. Since the sign bit of a reserved operand makes it look negative, copysign(x,reserved operand) = -x; should this return the reserved operand instead? 4.3 Berkeley Distribution May 12, 1986 IEEE(3M)

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