GNU Scientific Library Reference Manual - Third Edition (v1.12)by M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth, F. Rossi Paperback (6"x9"), 592 pages, 60 figures ISBN 0954612078 RRP £24.95 ($39.95) |

## 40.1 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.

ISBN 0954612078 | GNU Scientific Library Reference Manual - Third Edition (v1.12) | See the print edition |