## 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.