Texas Instruments TMS320C3x - Manual (Page 151)

Fast Logarithms on a Floating-Point Device
5-46
are equivalent to the seven MSBs of the logarithm. If the exponent could hold
all the bits needed for full accuracy, then it would be possible to continue the op-
eration for all 24 bits of the mantissa. Since there are only eight bits in the expo-
nent and the MSBs are used for negative values, only seven iterations are pos-
sible before the exponent must be off-loaded and reinitialized to zero.
By concatenating EXP_new to the previous exponent, longer strings of bits can
be built for greater accuracy. The process is then repeated until the desired accu-
racy is achieved. Also remember that the original numbers exponent, which rep-
resents the whole number part of the result, becomes the eight MSBs of the final
result.
Another technique is to look at the three MSBs of the mantissa and apply a
roundup from the fourth bit. Those same MSBs can be used as a quick exten-
sion of the exponent (logarithm). To visualize this, consider the following tabu-
lated values and graph.
1.0
0.585
0.5
0.0
1.0 1.5 2.0
Mantissa
Fraction
Mant log 2 (Mant)
1.000
1.250
1.500
1.7500
1.999 0.000
0.322
0.585
0.807
0.999
Figure 5–22. Tabulated Values for Mantissa
Note:
The fractional part is the same at the endpoints.
In the middle, only a slight bowing exists which can either be ignored or optionally
rounded for better accuracy. The maximum actually occurs at a mantissa value
of
1
ln(2.0)
or 1.442695. The value of log2(mant) at that point is 0.52876637, giving a maxi-
mum error of 0.086071.