Logic behind choosing the exponent bias as $2^7 -1$ instead of $2^7$ in $32$ bits IEEE-754 floating point representation
Jerome T. Coonen, &quot;An Implementation Guide to a Proposed Standard for Floating-Point Arithmetic,&quot; Computer, Vol. 13, No. 1, January 1980, pp. 68-79
provides the following rationale:

Because of the care taken in the treatment of Underflows, the range of
normalized numbers in single, double, and quad formats has been
chosen to diminish slightly the risk of Overflow compared with the
risk of Underflow. This was done by picking the exponent bias and
alignment of the binary point in the significant digit field in such a
way that the product of the largest and smallest positive normalized
numbers is roughly 4 in each of the basic formats.

David Stevenson, &quot;A Proposed Standard for Binary Floating-Point Arithmetic,&quot; Computer, Vol. 14, No. 3, March 1981, pp. 51-62
provides a somewhat different but related reasoning:

For the 32-bit format, precision was deemed the most important
criterion, hence the choice of radix 2 instead of octal or
hexadecimal. Other characteristics include not representing the
leading significand bit in normalized numbers, a minimally acceptable
exponent range which uses eight bits, and an exponent bias which
allows the reciprocal of all normalized numbers to be represented
without overflow.

David Goldberg, &quot;What Every Computer Scientist Should Know About Floating-Point Arithmetic&quot;, ACM Computing Surveys, Vol. 23, No. 1, March 1991, pp. 5-48
combines the above two explanations as follows:

Referring to Table 1, single precision has emax = 127 and
emin= -126. The reason for having |emin| &lt; emax is so that the reciprocal of the smallest number (1/2emin) will not overflow. Although it is
true that the reciprocal of the largest number will underflow,
underflow is usually less serious than overflow.