🚀 go-pugleaf

In article <9105130317.AA09926@ucbvax.Berkeley.EDU> jbs@WATSON.IBM.COM writes:
>
>          Herman Rubin states:
>                                              I have already pointed out
>that every trigonometric and exponential routine does, in some way,
>float/float -> integer, remainder.  The integer is also used.
>
>          1.  As I have already pointed out, the actual computation being
>done is float/real -> integer, remainder (since pi and ln(2) are not ma-
>chine numbers).  A proposed float/float -> integer, remainder instruc-
>tion (assuming it ran at the same speed as floating divide, actually it
>would probably be slower) would not benefit the trigonometric or
>exponential routines on any architecture I am familiar with

	Suppose we want sin(1234.0).  Approximate pi/2 = pi1 + pi2
where pi1 and pi2 are each double precision, | pi1 | >> | pi2 |. 
For purposes of illustration, let

		pi1 =  1.571
		pi2 = - .0002037

(decimal notation with four significant digits) so that 
pi1 + pi2 = 1.5707963 whereas pi/2 = 1.57079632679 ... .  Herman's
proposed instruction would find

		1234.0 = 785*pi1 + 0.765 .

If the quotient is always rounded to the nearest integer, then
the remainder (here 0.765) can be represented exactly even though 
785*1.571 = 1233.235 requires more that four significant digits.
How can Herman Rubin get this remainder accurately with the hardware 
and software commonly available? 

	jbs argues that this remainder 0.765 is only an approximation
to 1234.0 mod pi/2, since the divisor 3.142 was an approximation.
This is where pi2 can be used, together with the first quotient and remainder:

	   1234.0 - 785*(pi1 + pi2) = (1234.0 - 785*pi1) - 785*pi2
	= 0.7650 + 785*.0002037 = 0.7650 + .1599045 = 0.9249

(using 4-digit arithmetic).  The revised remainder of 0.9249 
compares to the true value 0.9248834 of 1234 - 785*(pi/2).  
With this we estimate sin(1234) ~= cos(0.9249) ~ 0.6019, 
which is accurate to four digits.
--
        Peter L. Montgomery 
        pmontgom@MATH.UCLA.EDU 
        Department of Mathematics, UCLA, Los Angeles, CA 90024-1555
If I spent as much time on my dissertation as I do reading news, I'd graduate.