@cindex IEEE floating point 

This chapter describes functions for examining the representation of
floating point numbers and controlling the floating point environment of
your program.  The functions described in this chapter are declared in
the header file @file{gsl_ieee_utils.h}.

@menu
* Representation of floating point numbers::  
* Setting up your IEEE environment::  
* IEEE References and Further Reading::  
@end menu

@node Representation of floating point numbers
@section Representation of floating point numbers
@cindex IEEE format for floating point numbers
@cindex bias, IEEE format
@cindex exponent, IEEE format
@cindex sign bit, IEEE format
@cindex mantissa, IEEE format
The IEEE Standard for Binary Floating-Point Arithmetic defines binary
formats for single and double precision numbers.  Each number is composed
of three parts: a @dfn{sign bit} (@math{s}), an @dfn{exponent}
(@math{E}) and a @dfn{fraction} (@math{f}).  The numerical value of the
combination @math{(s,E,f)} is given by the following formula,
@tex
\beforedisplay
$$
(-1)^s (1 \cdot fffff\dots) 2^E
$$
\afterdisplay
@end tex
@ifinfo

@example
(-1)^s (1.fffff...) 2^E
@end example

@end ifinfo
@noindent
@cindex normalized form, IEEE format
@cindex denormalized form, IEEE format
The sign bit is either zero or one.  The exponent ranges from a minimum value
@c{$E_{min}$}
@math{E_min} 
to a maximum value
@c{$E_{max}$}
@math{E_max} depending on the precision.  The exponent is converted to an 
unsigned number
@math{e}, known as the @dfn{biased exponent}, for storage by adding a
@dfn{bias} parameter,
@c{$e = E + \hbox{\it bias}$}
@math{e = E + bias}.
The sequence @math{fffff...} represents the digits of the binary
fraction @math{f}.  The binary digits are stored in @dfn{normalized
form}, by adjusting the exponent to give a leading digit of @math{1}. 
Since the leading digit is always 1 for normalized numbers it is
assumed implicitly and does not have to be stored.
Numbers smaller than 
@c{$2^{E_{min}}$}
@math{2^(E_min)}
are be stored in @dfn{denormalized form} with a leading zero,
@tex
\beforedisplay
$$
(-1)^s (0 \cdot fffff\dots) 2^{E_{min}}
$$
\afterdisplay
@end tex
@ifinfo

@example
(-1)^s (0.fffff...) 2^(E_min)
@end example

@end ifinfo
@noindent
@cindex zero, IEEE format
@cindex infinity, IEEE format
This allows gradual underflow down to 
@c{$2^{E_{min} - p}$}
@math{2^(E_min - p)} for @math{p} bits of precision. 
A zero is encoded with the special exponent of 
@c{$2^{E_{min}-1}$}
@math{2^(E_min - 1)} and infinities with the exponent of 
@c{$2^{E_{max}+1}$}
@math{2^(E_max + 1)}.

@noindent
@cindex single precision, IEEE format
The format for single precision numbers uses 32 bits divided in the
following way,

@smallexample
seeeeeeeefffffffffffffffffffffff
    
s = sign bit, 1 bit
e = exponent, 8 bits  (E_min=-126, E_max=127, bias=127)
f = fraction, 23 bits
@end smallexample

@noindent
@cindex double precision, IEEE format
The format for double precision numbers uses 64 bits divided in the
following way,

@smallexample
seeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffffff

s = sign bit, 1 bit
e = exponent, 11 bits  (E_min=-1022, E_max=1023, bias=1023)
f = fraction, 52 bits
@end smallexample

@noindent
It is often useful to be able to investigate the behavior of a
calculation at the bit-level and the library provides functions for
printing the IEEE representations in a human-readable form.

@comment float vs double vs long double 
@comment (how many digits are available for each)

@deftypefun void gsl_ieee_fprintf_float (FILE * @var{stream}, const float * @var{x})
@deftypefunx void gsl_ieee_fprintf_double (FILE * @var{stream}, const double * @var{x})
These functions output a formatted version of the IEEE floating-point
number pointed to by @var{x} to the stream @var{stream}. A pointer is
used to pass the number indirectly, to avoid any undesired promotion
from @code{float} to @code{double}.  The output takes one of the
following forms,

@table @code
@item NaN
the Not-a-Number symbol

@item Inf, -Inf
positive or negative infinity

@item 1.fffff...*2^E, -1.fffff...*2^E 
a normalized floating point number

@item 0.fffff...*2^E, -0.fffff...*2^E 
a denormalized floating point number

@item 0, -0
positive or negative zero

@comment @item [non-standard IEEE float], [non-standard IEEE double]
@comment an unrecognized encoding
@end table

The output can be used directly in GNU Emacs Calc mode by preceding it
with @code{2#} to indicate binary.
@end deftypefun

@deftypefun void gsl_ieee_printf_float (const float * @var{x})
@deftypefunx void gsl_ieee_printf_double (const double * @var{x})
These functions output a formatted version of the IEEE floating-point
number pointed to by @var{x} to the stream @code{stdout}.
@end deftypefun

@noindent
The following program demonstrates the use of the functions by printing
the single and double precision representations of the fraction
@math{1/3}.  For comparison the representation of the value promoted from
single to double precision is also printed.

@example
@verbatiminclude examples/ieee.c
@end example

@noindent
The binary representation of @math{1/3} is @math{0.01010101... }.  The
output below shows that the IEEE format normalizes this fraction to give
a leading digit of 1,

@smallexample
 f= 1.01010101010101010101011*2^-2
fd= 1.0101010101010101010101100000000000000000000000000000*2^-2
 d= 1.0101010101010101010101010101010101010101010101010101*2^-2
@end smallexample

@noindent
The output also shows that a single-precision number is promoted to
double-precision by adding zeros in the binary representation.

@comment importance of using 1.234L in long double calculations

@comment @example
@comment int main (void)
@comment @{
@comment   long double x = 1.0, y = 1.0;
  
@comment   x = x + 0.2;
@comment   y = y + 0.2L;

@comment   printf(" d %.20Lf\n",x);
@comment   printf("ld %.20Lf\n",y);

@comment   return 1;
@comment @}

@comment  d 1.20000000000000001110
@comment ld 1.20000000000000000004
@comment @end example


@node Setting up your IEEE environment
@section Setting up your IEEE environment
@cindex IEEE exceptions
@cindex precision, IEEE arithmetic
@cindex rounding mode
@cindex arithmetic exceptions
@cindex exceptions, IEEE arithmetic
@cindex division by zero, IEEE exceptions
@cindex underflow, IEEE exceptions
@cindex overflow, IEEE exceptions
The IEEE standard defines several @dfn{modes} for controlling the
behavior of floating point operations.  These modes specify the important
properties of computer arithmetic: the direction used for rounding (e.g.
whether numbers should be rounded up, down or to the nearest number),
the rounding precision and how the program should handle arithmetic
exceptions, such as division by zero.

Many of these features can now be controlled via standard functions such
as @code{fpsetround}, which should be used whenever they are available.
Unfortunately in the past there has been no universal API for
controlling their behavior---each system has had its own low-level way
of accessing them.  To help you write portable programs GSL allows you
to specify modes in a platform-independent way using the environment
variable @code{GSL_IEEE_MODE}.  The library then takes care of all the
necessary machine-specific initializations for you when you call the
function @code{gsl_ieee_env_setup}.

@deftypefun void gsl_ieee_env_setup ()
@vindex @env{GSL_IEEE_MODE}
This function reads the environment variable @code{GSL_IEEE_MODE} and
attempts to set up the corresponding specified IEEE modes.  The
environment variable should be a list of keywords, separated by
commas, like this,

@display
@code{GSL_IEEE_MODE} = "@var{keyword},@var{keyword},..."
@end display

@noindent
where @var{keyword} is one of the following mode-names,

@itemize @w{}
@item 
@code{single-precision}
@item 
@code{double-precision}
@item 
@code{extended-precision}
@item 
@code{round-to-nearest}
@item 
@code{round-down}
@item 
@code{round-up}
@item 
@code{round-to-zero}
@item 
@code{mask-all}
@item 
@code{mask-invalid}
@item 
@code{mask-denormalized}
@item 
@code{mask-division-by-zero}
@item 
@code{mask-overflow}
@item 
@code{mask-underflow}
@item 
@code{trap-inexact}
@item 
@code{trap-common}
@end itemize

If @code{GSL_IEEE_MODE} is empty or undefined then the function returns
immediately and no attempt is made to change the system's IEEE
mode.  When the modes from @code{GSL_IEEE_MODE} are turned on the
function prints a short message showing the new settings to remind you
that the results of the program will be affected.

If the requested modes are not supported by the platform being used then
the function calls the error handler and returns an error code of
@code{GSL_EUNSUP}.  

When options are specified using this method, the resulting mode is
based on a default setting of the highest available precision (double
precision or extended precision, depending on the platform) in
round-to-nearest mode, with all exceptions enabled apart from the
@sc{inexact} exception.  The @sc{inexact} exception is generated
whenever rounding occurs, so it must generally be disabled in typical
scientific calculations.  All other floating-point exceptions are
enabled by default, including underflows and the use of denormalized
numbers, for safety.  They can be disabled with the individual
@code{mask-} settings or together using @code{mask-all}.

The following adjusted combination of modes is convenient for many
purposes,

@example
GSL_IEEE_MODE="double-precision,"\
                "mask-underflow,"\
                  "mask-denormalized"
@end example

@noindent
This choice ignores any errors relating to small numbers (either
denormalized, or underflowing to zero) but traps overflows, division by
zero and invalid operations.

Note that on the x86 series of processors this function sets both the
original x87 mode and the newer @sc{mxcsr} mode, which controls SSE
floating-point operations.  The SSE floating-point units do not have a
precision-control bit, and always work in double-precision.  The
single-precision and extended-precision keywords have no effect in
this case.
@end deftypefun

@noindent
To demonstrate the effects of different rounding modes consider the
following program which computes @math{e}, the base of natural
logarithms, by summing a rapidly-decreasing series,
@tex
\beforedisplay
$$
e = 1 + {1 \over 2!} + {1 \over 3!} + {1 \over 4!} + \dots 
  = 2.71828182846...
$$
\afterdisplay
@end tex
@ifinfo

@example
e = 1 + 1/2! + 1/3! + 1/4! + ... 
  = 2.71828182846...
@end example
@end ifinfo

@example
@verbatiminclude examples/ieeeround.c
@end example

@noindent
Here are the results of running the program in @code{round-to-nearest}
mode.  This is the IEEE default so it isn't really necessary to specify
it here,

@example
$ GSL_IEEE_MODE="round-to-nearest" ./a.out 
i= 1 sum=1.000000000000000000 error=-1.71828
i= 2 sum=2.000000000000000000 error=-0.718282
....
i=18 sum=2.718281828459045535 error=4.44089e-16
i=19 sum=2.718281828459045535 error=4.44089e-16
@end example

@noindent
After nineteen terms the sum converges to within @c{$4 \times 10^{-16}$}
@math{4 \times 10^-16} of the correct value.  
If we now change the rounding mode to
@code{round-down} the final result is less accurate,

@example
$ GSL_IEEE_MODE="round-down" ./a.out 
i= 1 sum=1.000000000000000000 error=-1.71828
....
i=19 sum=2.718281828459041094 error=-3.9968e-15
@end example

@noindent
The result is about 
@c{$4 \times 10^{-15}$}
@math{4 \times 10^-15} 
below the correct value, an order of magnitude worse than the result
obtained in the @code{round-to-nearest} mode.

If we change to rounding mode to @code{round-up} then the final result
is higher than the correct value (when we add each term to the sum the
final result is always rounded up, which increases the sum by at least
one tick until the added term underflows to zero).  To avoid this
problem we would need to use a safer converge criterion, such as
@code{while (fabs(sum - oldsum) > epsilon)}, with a suitably chosen
value of epsilon.

Finally we can see the effect of computing the sum using
single-precision rounding, in the default @code{round-to-nearest}
mode.  In this case the program thinks it is still using double precision
numbers but the CPU rounds the result of each floating point operation
to single-precision accuracy.  This simulates the effect of writing the
program using single-precision @code{float} variables instead of
@code{double} variables.  The iteration stops after about half the number
of iterations and the final result is much less accurate,

@example
$ GSL_IEEE_MODE="single-precision" ./a.out 
....
i=12 sum=2.718281984329223633 error=1.5587e-07
@end example

@noindent
with an error of 
@c{$O(10^{-7})$}
@math{O(10^-7)}, which corresponds to single
precision accuracy (about 1 part in @math{10^7}).  Continuing the
iterations further does not decrease the error because all the
subsequent results are rounded to the same value.

@node IEEE References and Further Reading
@section References and Further Reading

The reference for the IEEE standard is,

@itemize @w{}
@item
ANSI/IEEE Std 754-1985, IEEE Standard for Binary Floating-Point Arithmetic.
@end itemize

@noindent
A more pedagogical introduction to the standard can be found in the
following paper,

@itemize @w{}
@item
David Goldberg: What Every Computer Scientist Should Know About
Floating-Point Arithmetic. @cite{ACM Computing Surveys}, Vol.@: 23, No.@: 1
(March 1991), pages 5--48.

Corrigendum: @cite{ACM Computing Surveys}, Vol.@: 23, No.@: 3 (September
1991), page 413. and see also the sections by B. A. Wichmann and Charles
B. Dunham in Surveyor's Forum: ``What Every Computer Scientist Should
Know About Floating-Point Arithmetic''. @cite{ACM Computing Surveys},
Vol.@: 24, No.@: 3 (September 1992), page 319.
@end itemize

@noindent

A detailed textbook on IEEE arithmetic and its practical use is
available from SIAM Press,

@itemize @w{}
@item
Michael L. Overton, @cite{Numerical Computing with IEEE Floating Point Arithmetic},
SIAM Press, ISBN 0898715717.
@end itemize

@noindent

@comment to turn on math exception handling use __setfpucw, see
@comment /usr/include/i386/fpu_control.h
@comment e.g.
@comment #include <math.h>
@comment #include <stdio.h>
@comment #include <fpu_control.h>
@comment double f (double x);
@comment int main ()
@comment {
@comment   double a = 0;
@comment   double y, z;
@comment   __setfpucw(0x1372); 
@comment mention extended vs double precision on Pentium, and how to get around
@comment it (-ffloat-store, or selective use of volatile)

@comment On the alpha the option -mieee is needed with gcc
@comment In Digital's compiler the equivalent is -ieee, -ieee-with-no-inexact 
@comment and -ieee-with-inexact,  or -fpe1 or -fpe2