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Decimal arithmetic, compatible with that used in Java, C#, PL/I, COBOL, Python, REXX, etc., is also defined in this section. In general, decimal arithmetic follows the same rules as binary arithmetic (results are correctly rounded, and so on), with additional rules that define the exponent of a result (more than one is possible in many cases).
convert a double to a float: d2i 8e 1000 1110 value → result convert a double to an int d2l 8f 1000 1111 value → result convert a double to a long dadd 63 0110 0011 value1, value2 → result add two doubles daload 31 0011 0001 arrayref, index → value load a double from an array dastore 52 0101 0010 arrayref, index, value →
The original binary value will be preserved by converting to decimal and back again using: [58] 5 decimal digits for binary16, 9 decimal digits for binary32, 17 decimal digits for binary64, 36 decimal digits for binary128. For other binary formats, the required number of decimal digits is [h]
Conversion of the fractional part: Consider 0.375, the fractional part of 12.375. To convert it into a binary fraction, multiply the fraction by 2, take the integer part and repeat with the new fraction by 2 until a fraction of zero is found or until the precision limit is reached which is 23 fraction digits for IEEE 754 binary32 format.
Thus, only 10 bits of the significand appear in the memory format but the total precision is 11 bits. In IEEE 754 parlance, there are 10 bits of significand, but there are 11 bits of significand precision (log 10 (2 11) ≈ 3.311 decimal digits, or 4 digits ± slightly less than 5 units in the last place).
The use of decimal when talking about binary is unfortunate because most decimal fractions are recurring sequences in binary just as 2 / 3 is in decimal. Thus, a value such as 10.15, is represented in binary as equivalent to 10.1499996185 etc. in decimal for REAL*4 but 10.15000000000000035527 etc. in REAL*8: inter-conversion will ...
Many non-integral values, such as decimal 0.2, have an infinite place-value representation in binary (.001100110011...) but have a finite place-value in binary-coded decimal (0.0010). Consequently, a system based on binary-coded decimal representations of decimal fractions avoids errors representing and calculating such values.
The decimal number 0.15625 10 represented in binary is 0.00101 2 (that is, 1/8 + 1/32). (Subscripts indicate the number base .) Analogous to scientific notation , where numbers are written to have a single non-zero digit to the left of the decimal point, we rewrite this number so it has a single 1 bit to the left of the "binary point".