  Recent evaluations have highlighted the tapered posit number format as a
promising alternative to the uniform precision IEEE 754 floating-point numbers,
which suffer from various deficiencies. Although the posit encoding scheme
offers superior coding efficiency at values close to unity, its efficiency
markedly diminishes with deviation from unity. This reduction in efficiency
leads to suboptimal encodings and a consequent diminution in dynamic range,
thereby rendering posits suboptimal for general-purpose computer arithmetic.
  This paper introduces and formally proves 'takum' as a novel general-purpose
logarithmic tapered-precision number format, synthesising the advantages of
posits in low-bit applications with high encoding efficiency for numbers
distant from unity. Takums exhibit an asymptotically constant dynamic range in
terms of bit string length, which is delineated in the paper to be suitable for
a general-purpose number format. It is demonstrated that takums either match or
surpass existing alternatives. Moreover, takums address several issues
previously identified in posits while unveiling novel and beneficial arithmetic
properties.
