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IEEE floating-point arithmetic

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 `gsl_ieee_utils.h'.

Representation of floating point numbers

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 sign bit (s), an exponent (E) and a fraction (f). The numerical value of the combination (s,E,f) is given by the following formula,

(-1)^s (1.fffff...) 2^E

The sign bit is either zero or one. The exponent ranges from a minimum value E_min to a maximum value E_max depending on the precision. The exponent is converted to an unsigned number e, known as the biased exponent, for storage by adding a bias parameter, e = E + bias. The sequence fffff... represents the digits of the binary fraction f. The binary digits are stored in normalized form, by adjusting the exponent to give a leading digit of 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 2^(E_min) are be stored in denormalized form with a leading zero,

(-1)^s (0.fffff...) 2^(E_min)

This allows gradual underflow down to 2^(E_min - p) for p bits of precision. A zero is encoded with the special exponent of 2^(E_min - 1) and infinities with the exponent of 2^(E_max + 1).

The format for single precision numbers uses 32 bits divided in the following way,

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

The format for double precision numbers uses 64 bits divided in the following way,

seeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffffff

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

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.

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

NaN
the Not-a-Number symbol
Inf, -Inf
positive or negative infinity
1.fffff...*2^E, -1.fffff...*2^E
a normalized floating point number
0.fffff...*2^E, -0.fffff...*2^E
a denormalized floating point number
0, -0
positive or negative zero

The output can be used directly in GNU Emacs Calc mode by preceding it with 2# to indicate binary.

Function: void gsl_ieee_printf_float (const float * x)
Function: void gsl_ieee_printf_double (const double * x)
These functions output a formatted version of the IEEE floating-point number pointed to by x to the stream stdout.

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

#include <stdio.h>
#include <gsl/gsl_ieee_utils.h>

int
main (void) 
{
  float f = 1.0/3.0;
  double d = 1.0/3.0;

  double fd = f; /* promote from float to double */
  
  printf (" f="); gsl_ieee_printf_float(&f); 
  printf ("\n");

  printf ("fd="); gsl_ieee_printf_double(&fd); 
  printf ("\n");

  printf (" d="); gsl_ieee_printf_double(&d); 
  printf ("\n");

  return 0;
}

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

 f= 1.01010101010101010101011*2^-2
fd= 1.0101010101010101010101100000000000000000000000000000*2^-2
 d= 1.0101010101010101010101010101010101010101010101010101*2^-2

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

Setting up your IEEE environment

The IEEE standard defines several 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 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 GSL_IEEE_MODE. The library then takes care of all the necessary machine-specific initializations for you when you call the function gsl_ieee_env_setup.

Function: void gsl_ieee_env_setup ()
This function reads the environment variable 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,

GSL_IEEE_MODE = "keyword,keyword,..."

where keyword is one of the following mode-names,

If 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 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 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 INEXACT exception. The 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 mask- settings or together using mask-all.

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

GSL_IEEE_MODE="double-precision,"\
                "mask-underflow,"\
                  "mask-denormalized"

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 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.

To demonstrate the effects of different rounding modes consider the following program which computes e, the base of natural logarithms, by summing a rapidly-decreasing series,

e = 1 + 1/2! + 1/3! + 1/4! + ... 
  = 2.71828182846...
#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_ieee_utils.h>

int
main (void)
{
  double x = 1, oldsum = 0, sum = 0; 
  int i = 0;

  gsl_ieee_env_setup (); /* read GSL_IEEE_MODE */

  do 
    {
      i++;
      
      oldsum = sum;
      sum += x;
      x = x / i;
      
      printf ("i=%2d sum=%.18f error=%g\n",
              i, sum, sum - M_E);

      if (i > 30)
         break;
    }  
  while (sum != oldsum);

  return 0;
}

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

$ 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

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

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

The result is about 4 \times 10^-15 below the correct value, an order of magnitude worse than the result obtained in the round-to-nearest mode.

If we change to rounding mode to 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 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 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 float variables instead of double variables. The iteration stops after about half the number of iterations and the final result is much less accurate,

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

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

References and Further Reading

The reference for the IEEE standard is,

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

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


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