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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 way of accessing them. For example, the Linux kernel provides the function `__setfpucw` (set-fpu-control-word) to set IEEE modes, while HP-UX and Solaris use the functions `fpsetround` and `fpsetmask`. 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,

  - `single-precision`
  - `double-precision`
  - `extended-precision`
  - `round-to-nearest`
  - `round-down`
  - `round-up`
  - `round-to-zero`
  - `mask-all`
  - `mask-invalid`
  - `mask-denormalized`
  - `mask-division-by-zero`
  - `mask-overflow`
  - `mask-underflow`
  - `trap-inexact`
  - `trap-common`

  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`.

  The following 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.

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 <math.h>
    #include <stdio.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 series no longer converges (the reason is that when we add each term to the sum the final result is always rounded up. This is guaranteed to increase the sum by at least one tick on each iteration). 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,

- ANSI/IEEE Std 754-1985, IEEE Standard for Binary Floating-Point Arithmetic

A more pedagogical introduction to the standard can be found in the paper "What Every Computer Scientist Should Know About Floating-Point Arithmetic".

- David Goldberg: What Every Computer Scientist Should Know About Floating-Point Arithmetic. ACM Computing Surveys, Vol. 23, No. 1 (March 1991), pages 5-48 Corrigendum: ACM Computing Surveys, Vol. 23, No. 3 (September 1991), page 413.
- 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". ACM Computing Surveys, Vol. 24, No. 3 (September 1992), page 319

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