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added sign 1 is negative, degrading precision for denormals, 'integral decoding' |
→IEEE 754 quadruple-precision binary floating-point format: binary128: added some details; use standard terminology. |
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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. If an IEEE 754 quadruple-precision number is converted to a decimal string with at least 36 significant digits, and then converted back to quadruple-precision representation, the final result must match the original number.<ref name="whyieee">{{cite web |author=Kahan |first=Wiliam |date=1 October 1987 |title=Lecture Notes on the Status of IEEE Standard 754 for Binary Floating-Point Arithmetic |url=http://www.cs.berkeley.edu/~wkahan/ieee754status/IEEE754.PDF}}</ref>
The format is written with an implicit lead bit with value 1 unless the exponent is stored with all zeros (used to encode [[subnormal number]]s and zeros). Thus only 112 bits of the [[significand]] appear in the memory format, but the total precision is 113 bits (approximately 34 decimal digits: {{nowrap|log<sub>10</sub>(2<sup>113</sup>) ≈ 34.016}}) for normal values
[[File:IEEE 754 Quadruple Floating Point Format.svg|800px|A sign bit, a 15-bit exponent, and a 112-bit significand]]
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