EVALUATION
We are aware of the issue raised by this bug; it is the same issue
raised in 4807358, the "almabench" bug. Both this bug and 4807358 are
symptoms of the same problem: the sin/cos routines in the client
compiler under 1.3.1 did *not* comply with the Math.{sin/cos}
specification. As explained below, the way they did not comply could
be the source of a large amount of error. As a consequence of fixing
this compliance problem (bug 4345903), the x86 sin/cos routines in 1.4
and later are slower than in 1.3.1 client.
The x87 FPU has fsin and fcos instructions to accelerate the
computation of sine and cosine. However, these instructions have a
number of limitations. First, they only return sensible results over a
limited range of values, +/ 2^63. Java's sin/cos functions are defined
over the full double range, roughly +/ 2^1023. Second, even within
the +/2^63 range, nearly all fsin/fcos implementations perform faulty
*argument reduction,* consequently, the results can be very wrong
outside of a narrow range of +/ pi/4.
The way many math libraries routines work is to map any argument into
an "equivalent" argument in a fixed, narrow range; this process is
called argument reduction. Since sine and cosine are periodic, the
basic idea behind the mapping is clear; subtract out multiples of
2*pi. The actual details are a little more sophisticated; in the case
of sin/cos, arguments are mapped into [pi/4, pi/4] using sign
symmetries between sin/cos (for a good current reference on such
matters see JeanMichel Muller's book "Elementary Functions").
The number pi is transcendental; it has no finite binary
representation. Around 1982 several years after the original x87
design, techniques for doing argument reduction as if by an exact
value of pi were developed (see http://www.validlab.com/arg.pdf for a
discussion of the techniques). The fsin/fcos instructions use an
older technique: pretend pi has a fixed number of bits; most x87
implementations use a 66bit approximation of pi for argument
reduction. If the sine/cosine functions are viewed a spring, using a
limited precision approximation to pi amounts to slightly compressing
the spring. Even for moderately large arguments, the difference
between the compressed and uncompressed versions of sin/cosine can be
very, very large, affecting even the sign of the result.
People are often unnerved when they discover that 1.0/10.0 cannot be
exactly represented as a binary floatingpoint number. The exact
value of the floatingpoint result of dividing 1 by 10 is
0.1000000000000000055511151231...
which is accurate to 17 decimal places. In floatingpoint parlance,
this is accurate to 1/2 ulp (unit in the last place), which is the
most one can ask from a single floatingpoint operation.
The value of sine for the floatingpoint number Math.PI is around
1.2246467991473532E16
while the computed value for the algorithm used in java 1.3.1 is
1.2246063538223773E16
^
In other words, the returned result is only accurate to about 5 digits
instead of the full 1517 digit precision of double. Instead of a 1/2
ulp or 1 ulp error, the error is about 1.64e11 ulps, over *ten
billion* ulps.
Does this magnitude of error matter to your application? It might, or
it might not. However, in case it does matter, the Java specification
requires an accurate answer be returned. If you don't care what is
returned, why do you care how fast it is returned? Rhetorically, sin
could always return 0 and cos could always return 1; they would be
very fast but not too useful as sine and cosine functions. (Neither
the almabench code nor the code submitted in this bug actually examine
the results to verify they are sensible.)
On a related note, the sine and cosine of large values, even as large
as Double.MAX_VALUE, are perfectly welldefined mathematically. There
is no mathematical justification for returning arbitrary sin/cos
results for large values.
In 1.4, an effort was made to take advantage of the fsin/fcos hardware
as much as we could and still meet the spec. However, "large"
arguments outside of around pi/4 require a separate argument reduction
step, which can be the bulk of the cost.
Closing as not a bug.
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