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Real Inferno
.AU
.I "Eric Grosse"
.AI
.I "Lucent Technologies, Bell Labs"
.I "Murray Hill NJ 07974 USA"
.I "ehg@bell-labs.com"
.\"date{19 Aug 1996, minor revisions 7 Jan 1998}
.FS
Previously appeared in R.F. Boisvert (editor),
.ft I
The Quality of Numerical Software: Assessment and
Enhancement: Proceedings of the IFIP TC2/WG2.5
Working Conference on the Quality of Numerical
Software, Oxford,
United Kingdom, 8-12 July 1996,
.ft R
Chapman Hall,
London,
1997 (pp. 270-279).
.FE
.AB
Inferno is an operating system well suited to applications that need to be
portable, graphical, and networked. This paper describes the fundamental
floating point facilities of the system, including: tight rules on
expression evaluation, binary/decimal conversion, exceptions and rounding,
and the elementary function library.
.AE
.PP
Although the focus of Inferno is interactive media, its portability across
hardware and operating platforms, its relative simplicity, and its strength
in distributed computing make it attractive for advanced scientific
computing as well. Since the appearance of a new operating system is a
relatively uncommon event, this is a special opportunity for numerical
analysts to voice their opinion about what fundamental facilities they need.
The purpose of this short paper is to describe numerical aspects of the
initial release of Inferno, and to invite comment before the tyranny of
backward compatibility makes changes impossible.
.PP
An overview of Inferno is given by Dorward et. al.
.]C "Inferno" ,
but for our immediate purposes it may suffice to say that Inferno plays the
role of a traditional operating system (with compilers, process control,
networking, graphics, and so on) but can run either on bare hardware or on
top of another operating system like Windows95 or Unix. Programs for
.I "Inferno"
are written in the language
.I "Limbo"
and compiled to
machine-independent object files for the
.I "Dis"
virtual
machine, which is then implemented with runtime compilation for best
performance. Files are accessible over networks using the
.I "Styx"
protocol; together with the presentation of most system resources as files
and the manipulation of file namespaces, this permits integration of a
collection of machines into a team. Limbo looks somewhat like a mixture of C
and Pascal, augmented by modules (to cope with the namespace and dynamic
loading needs of large programs) and by a channel facility for convenient
(coarse-grain) parallel programing. Array references are bounds-checked and
memory is garbage collected.
.PP
The rest of this paper covers the fundamental floating point environment
provided by the Limbo compiler and
.I "math"
module, the ``elementary
functions,'' and finally some comments on why particular definitions were
chosen or why certain facilities were included or excluded. This discussion
assumes the reader is familiar with scientific computing in general and the
IEEE floating point standard in particular.
.NH 1
Floating point
.PP
In Limbo, arithmetic on literal and named constants is evaluated at compile
time with all exceptions ignored. Arithmetic on variables is left by the
compiler to runtime, even if data path analysis shows the value to be a
compile time constant. This implies that tools generating Limbo source must
do their own simplification, and not expect the compiler to change $x/x$
into $1$, or $-(y-x)$ into $x-y$, or even $x-0$ into $x$. Negation $-x$
changes the sign of $x$; note that this not the same as $0-x$ if $x=0$.
.PP
The compiler may perform subexpression elimination and other forms of code
motion, but not across calls to the mode and status functions. It respects
parentheses. The evaluation order of $a+b+c$ follows the parse tree and is
therefore the same as for $(a+b)+c$. These are the same rules as for Fortran
and C.
.PP
Contracted multiply-add instructions (with a single rounding) are not
generated by the compiler, though they may be used in the native
.SM BLAS
libraries. All arithmetic follows the IEEE floating point standard
.]C "IEEEfp" ,
except that denormalized numbers may not be supported; see the
discussion in section 3.
.PP
The most important numerical development at the language level recently has
been accurate binary/decimal conversion
.]C "Clinger"
.]C "Gay"
.]C "SteeleWhite" .
Thus printing a real using
.CW "\%g"
and reading
it on a different machine guarantees recovering identical bits. (Limbo uses
the familiar
.I "printf"
syntax of C, but checks argument types against
the format string at compile time, in keeping with its attempt to help the
programmer by stringent type checking.) A good
.I "strtod/dtoa"
is,
unfortunately, 1700 lines of source (15kbytes compiled), though with modest
average runtime penalty. This code must be used in the compiler so that
coefficients are accurately transferred to bytecodes. Smaller, faster, but
sloppier, runtimes will also be provided when mandated by limited memory and
specialized use. However, programmers may assume the features described in
this paper are present in all Inferno systems intended for general computing.
.PP
Each thread has a floating point control word (governing rounding mode and
whether a particular floating point exception causes a trap) and a floating
point status word (storing accumulated exception flags). Functions
.I "FPcontrol"
and
.I "FPstatus"
copy bits to the control or status word, in
positions specified by a mask, returning previous values of the bits.
.I "getFPcontrol"
and
.I "getFPstatus"
return the words unchanged.
.PP
The constants
.I "INVAL, ZDIV, OVFL, UNFL, INEX"
are non-overlapping
single-bit masks used to compose arguments or return values. They stand for
the five IEEE exceptions:
.IP •
``invalid operation'' ($0/0$,$0 * infinity $,$ infinity - infinity $,$sqrt{-1}$)
.IP •
``division by zero'' ($1/0$),
.IP •
``overflow'' ($1.8e308$)
.IP •
``underflow'' ($1.1e-308$)
.IP •
``inexact'' ($.3*.3$).
.PP
The constants
.I "RND_NR, RND_NINF, RND_PINF, RND_Z"
are distinct
bit patterns for ``round to nearest even'', ``round toward $-{infinity} $'',
``round toward $+{infinity} $'', ``round toward $0$'', any of which can be set
or extracted from the floating point control word using
.I "RND_MASK" .
For example,
.IP •
to arrange for the program to tolerate underflow,
.I "FPcontrol(0,UNFL)."
.IP •
to check and clear the inexact flag,
.I "FPstatus(0,INEX)."
.IP •
to set directed rounding,
.I "FPcontrol(RND_PINF,RND_MASK)."
.PP
By default,
.I "INEX"
is quiet and
.I "OVFL, UNFL, ZDIV,"
and
.I "INVAL"
are fatal. By default, rounding is to nearest even, and library
functions are entitled to assume this. Functions that wish to use quiet
overflow, underflow, or zero-divide should either set and restore the
control register themselves or clearly document that the caller must do so.
The ``default'' mentioned here is what a Limbo program gets if started in a
fresh environment. Threads inherit floating point control and status from
their parent at the time of spawning and therefore one can spawn a ``round
toward 0'' shell and re-run a program to effortlessly look for rounding
instabilities in a program.
.NH 1
Elementary functions
.PP
The constants
.I "Infinity, NaN, MachEps, Pi, Degree"
are defined. Since
Inferno has thorough support of Unicode, it was tempting to name these $infinity $, $ε $, $π $, and °, but people (or rather, their
favorite text editing tools) may not be ready yet for non-\s-2ASCII\s0
source text.
.I "Infinity"
and
.I "NaN"
are the positive infinity
and quiet not-a-number of the IEEE standard, double precision.
.I MachEps
is $2 sup {-52}$, the unit in the last place of the mantissa $1.0$.
The value of
.I "Pi"
is the nearest machine number to the
mathematical value $π $.
.I "Degree"
is
$"Pi" / 180$.
.PP
Three useful functions
.I "fdim, fmax, fmin"
are adopted from the
Numerical C extensions
.]C "NumerC" .
The unusual one of these, often
denoted $(x-y) sub {+}$, is defined by $roman "fdim" ( x , y )=x-y$ if $x > y$, else $0$. The compiler may turn these into efficient machine instruction sequences,
possibly even branch-free, rather than function calls. There are two almost
redundant mod functions:
.I "remainder(x,y)"
is as defined by the IEEE
standard (with result $roman "|" r roman "|" <= y/2$);
.I "fmod(x,y)"
is $x roman "mod" y$,
computed in exact arithmetic with $0<= r<y$. Limbo has a ``tuple'' type,
which is the natural return value in the call $(i,f)= roman "modf" ( x )$ to
break $x$ into integer and fractional parts. The function
.I "rint"
rounds to an integer, following the rounding mode specified in the floating
point control word.
.PP
For a good-quality, freely-available elementary function library,
.I "math"
uses the IEEE subset of
.I "fdlibm"
.]C "fdlibm" .
Of course, a
conforming implementation may use entirely different source, but must take
care with accuracy and with special arguments. There are the customary
power, trigonometric, Bessel, and erf functions, and specialized versions $roman "expm1"( x )=e sup x - 1$, $roman "log1p" ( x )=log ( 1 + x )$. An additional function
$roman "pow10"( n ) = 10 sup n$ is defined; in the default implementation this is
just fdlibm's $roman "pow" ( 10. , n )$ but it is provided so that separate
trade-offs of accuracy and simplicity can be made
.]C "MacIlroy" .
.I "fdlibm"
uses extra precise argument reduction, so the computed $sin (n*Pi)$
is small but nonzero. If demands warrant, degree versions of the
trigonometric functions will be added, but for now the style $sin (45*Degree)$ is used.
The library also provides IEEE functions
.I "ilogb, scalbn, copysign, finite, isnan,"
and
.I "nextafter" .
.PP
The functions
.I "dot, norm1, norm2, iamax, gemm"
are adopted from the
.SM BLAS
.]C "blas"
to get tuned linear algebra kernels for
each architecture, possibly using extra-precise accumulators. These are
defined by $sum {{x sub i}{y sub i}}$, $sum roman | {x sub i} roman | $, $ sqrt{sum { x sub {i sup 2}}} $, $i$ such
that $roman | {x sub i} roman | = roman max $, and $C= alpha AB + beta C$ with optional transposes on $A$
and $B$. Since Limbo has only one floating-point type, there is no need here
for a precision prefix. Limbo array slices permit the calling sequences to
be more readable than in Fortran77 or C, though restricted to unit stride.
This encourages better cache performance anyway. The matrix multiply
function
.I "gemm"
remains general stride (and is the foundation for
other operations
.]C "Kagstrom" ).
.PP
Limbo is like C in providing singly-subscripted arrays with indexing
starting at 0. Although Limbo offers arrays of arrays, as in C, for
scientific work a better choice is to adopt the style of linearizing
subscripts using Fortran storage order. This promotes easier exchange of
data with other applications and reuses effort in organizing loops to
achieve good locality. In previous language work
.]C "pine" ,
we implemented
a C preprocessor that allowed the programmer to choose a convenient origin
(such as 1) and have it compiled into 0 for the base language; because we
passed arrays as dope vectors, we were even able to allow different origins
for the same array in calling and called functions. The main lesson we
learned from that experience, however, was that permutations become a
nightmare when there is anything but dogmatic adherence to a single origin.
So for an $m$ by $n$ matrix $A$, the programmer should use loops with $0<=
i<m$ and $0<= j<n$ and access $A[i+m*j]$.
.PP
For interoperability with foreign file formats and for saving main memory in
selected applications, functions are provided for copying bits between and
reals and 32-bit or 64-bit IEEE-format values.
.PP
Finally,
.I "math"
provides a tuned quicksort function
.I "sort(x,p)"
where
.I "x"
is a real array and
.I "p"
is an int array representing
a 0-origin permutation. This function leaves the contents of
.I "x"
untouched and rearranges
.I "p"
so that $x[{p sub i}]<= x[p sub {i+1}]$. This is
usually what one wants to do: sort an array of abstract data types based on
some key, but without the need to actually swap large chunks of memory.
.NH 1
Rationale
.PP
This section discusses why certain numerical features were included or not.
.NH 2
Rounding modes and accumulated exceptions
.PP
Directed rounding is only needed in a very few places in scientific
computing, but in those places it is indispensable. Accumulated floating
point exceptions are even more useful. User trap handling is a harder
problem, and may be worth leaving for later, possibly with a default
``retrospective diagnostics'' log
.]C "Kahan" .
.PP
Note that the exception masks must be architecture independent, since they
reside in the Limbo bytecodes, and therefore the implementation involves a
small amount of bit fiddling. Still, it is efficient enough to encourage
use. It would be difficult to port to a processor that only had static
rounding modes in instruction opcodes rather than the dynamic model
specified in section 2 of the IEEE standard. Fortunately, the Alpha
does provide both models.
.NH 2
Sudden underflow
.PP
Some processor vendors make supporting gradual underflow just too hard. (One
must struggle upon the system trap to reconstruct exactly which instruction
was executing and what the state of the registers was. On the MIPS, it is
said to be 30 pages of assembler.) So Inferno supports denormalized numbers
only if the hardware makes this easy. Providing underflow that is correct
but very slow, as some systems do, is not necessarily doing the user a favor.
.PP
To determine portably if a particular system offers gradual underflow, mask
off UNFL and do trial arithmetic.
.NH 2
Speed
.PP
Computers with slow (software) gradual underflow usually provide a fast
flush-to-0 alternative. This often suffices, though there are important
examples where it forces an uglier and slower coding style. A worse
situation is if the hardware uses system traps for Infinity and NaN
arithmetic. The resulting slowdown will make otherwise excellent and natural
algorithms run slowly
.]C "Demmel" .
Sadly, even some x86 implementations
that do non-finite arithmetic in hardware, do it relatively slowly.
.PP
We considered providing syntax to declare a certain program scope within
which precise IEEE behavior was required, and relaxing the rules outside
such scopes.
(The numerical C extensions
.]C "NumerC"
use pragma
for this purpose.)
These scope declarations would need to be in the
bytecodes, since significant optimization may be attempted by the runtime
compiler. After some discussion, and with some trepidation, it was agreed
that instead all compilers would be required to preserve the same result and
status as for an unoptimized version.
.NH 2
Comparison
.PP
The standard C operators
.CW ==
.CW !=
.CW "<"
.CW "<="
.CW ">"
.CW ">="
are the only comparisons provided, and they behave exactly
like the ``math'' part of Table 4 of the IEEE standard. Programs interested
in handling NaN data should test explicitly. This seems to be the way most
people program and leads to code more understandable to nonexperts. It is
true that with more operators one can correctly write code that propagates
NaNs to a successful conclusion\-but that support has been left for later.
NaN(''tag'') should be added at that same time.
.NH 2
Precision
.PP
All implementations run exclusively in IEEE double precision. If the
hardware has extra-precise accumulators, the round-to-double mode is set
automatically and not changeable, in keeping with Limbo's design to have
only one floating point type. Extended precision hardware, if available, may
be used by the built-in elementary function and
.SM BLAS
libraries.
Also, we contemplate adding a dotsharp function that would use a very long
accumulator for very precise inner products, independent of the order of
vector elements
.]C "kulisch" .
But reference implementations that use only
double precision, avoid contracted multiply-add, and evaluate in the order 1
up to n will always be available for strict portability.
.PP
At the time the decision was made to restrict the system to 64-bit floating
point, Limbo integers were almost exclusively 32-bit and the consistency
argument to have a single real type was compelling. Now that Limbo has more
integer types the decision might be reconsidered. But so many engineers
needlessly struggle with programs run in short precision, that offering it
may do as much harm as good. On most modern computers used for general
purpose scientific computing, 64-bit floating point arithmetic is as fast as
32-bit, except for the memory traffic. In cases where the shorter precision
would suffice and memory is a crucial concern, the programmer should
consider carefully scaled fixed point or specialized compression. To
efficiently interoperate with data files that use the short format,
programmers may use the provided realbits32 function. While there are surely
appropriate uses for a first-class 32-bit real type, for now we follow
Kahan's sarcastic motto ``why use lead when gold will do?''
.NH 2
BLAS
.PP
The few
.SM BLAS
in the core library were chosen for readability and,
in case of gemm, for optimization beyond what a reasonable compiler would
attempt. We expect that compilers will (soon) be good enough that the
difference between compiling $y+=a*x$ and calling daxpy is small. Also, as
mentioned above, dot and gemm might reasonably use combined multiply-add or
a long accumulator in some optional implementations.
.NH 2
$GAMMA ( x )$
.PP
To avoid confusion with the C math library, which defined
.I "gamma"
as $ln GAMMA $, we offer only
.I "lgamma"
for now. This function and
.I "modf"
return an (int,real) tuple rather than assigning through an
integer pointer, in keeping with Limbo's design. The opportunity has been
taken to drop some obsolete functions like
.I "frexp" .
Other functions
are unchanged from the C math library.
.NH 2
Future
.PP
A prototype preprocessor has been written to allow the scientific programmer
to write $A[i,j]$ for an $A$ that was created as a $Matrix(m,n)$ and to have
the subscript linearization done automatically. Here $Matrix$ is an Limbo
abstract data type containing a real array and integers $m$, $n$, and column
stride $lda$ used as in typical Fortran calling sequences.
.PP
The Limbo compiler is soon expected to implement the type
.I "complex" .
.PP
Higher level numerical libraries will also be provided, and although that
topic is beyond the scope of this paper, opinions about what should come
first would be welcome.
.PP
Distributed computing has not been mentioned here because it involves
relatively few considerations specific to floating point computation.
However, it may be worth noting that in the default environment (with
underflow trapped, so that presence or absence of denormalized numbers is
not significant) programs run independently on heterogeneous machines
nevertheless get precisely identical results, even with respect to thread
scheduling. This implies that certain communication steps can be avoided,
and that regression testing is considerably simplified.
.PP
Please direct comments on these numerical aspects of Inferno to Eric Grosse.
More general technical comments can be directed to Vita Nuova
.CW comments@vitanuova.com ). (
I am grateful to Joe Darcy, Berkeley,
to David Gay, Bell Labs, to David Hook, University of Melbourne,
and to participants of the IFIP WG2.5 Working
Conference on Quality of Numerical Software for insightful comments on a
first draft of this paper.
.\"the principal developers of Inferno: Sean Dorward, Rob Pike, Dave Presotto, Howard Trickey, and Phil Winterbottom.
.SH
Trademarks
.LP
Inferno, Limbo, and Dis are trademarks of Vita Nuova Holdings Limited.
Unix is a trademark of Unix Systems Laboratories.
Windows95 is a trademark of Microsoft.
.EQ
delim off
.EN
.SH
References
.nr PS -1
.nr VS -1
.LP
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