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A fixed-point representation of a fractional number is essentially an integer that is to be implicitly multiplied by a fixed scaling factor. For example, the value 1.23 can be stored in a variable as the integer value 1230 with implicit scaling factor of 1/1000 (meaning that the last 3 decimal digits are implicitly assumed to be a decimal fraction), and the value 1 230 000 can be represented ...
Thus only 23 fraction bits of the significand appear in the memory format, but the total precision is 24 bits (equivalent to log 10 (2 24) ≈ 7.225 decimal digits) for normal values; subnormals have gracefully degrading precision down to 1 bit for the smallest non-zero value.
Be aware that the bit numbering used here for e.g. b 9 … b 0 is in opposite direction than that used in the document for the IEEE 754 standard b 0 … b 9, add. the decimal digits are numbered 0-base here while in opposite direction and 1-based in the IEEE 754 paper. The bits on white background are not counting for the value, but signal how ...
For example, 0.1 in decimal (1/10) is 0b1/0b1010 in binary, by dividing this in that radix, the result is 0b0.0 0011 (because one of the prime factors of 10 is 5). For more general fractions and bases see the algorithm for positive bases.
In contrast to the binaryxxx data formats the decimalxxx formats provide exact representation of decimal fractions, exact calculations with them and enable human common 'ties away from zero' rounding [1] (in some range, to some precision, to some degree). In a trade-off for reduced performance, which is especially harming decimal128 ...
- Different understanding of significand as integer or fraction, and acc. different bias to apply for the exponent (for decimal64 what is stored in bits can be decoded as base to the power of 'stored value for the exponent minus bias of 383' times significand understood as d 0. d −1 d −2 d −3 d −4 d −5 d −6 d −7 d −8 d −9 d ...
This gives from 33 to 36 significant decimal digits precision. If a decimal string with at most 33 significant digits is converted to the IEEE 754 quadruple-precision format, giving a normal number, and then converted back to a decimal string with the same number of digits, the final result should match the original string.
The only positive integers that can be non-Brazilian are 1, 6, the primes, and the squares of the primes, for every other number is the product of two factors x and y with 1 < x < y − 1, and can be written as xx in base y − 1. [14] If a square of a prime p 2 is Brazilian, then prime p must satisfy the Diophantine equation