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--- pdclib:printing_floating_point_numbers [2025/08/21 12:49] – solar
+++ pdclib:printing_floating_point_numbers [2025/08/21 14:01] (current) – [Biased Exponent] solar
@@ Line 25: / Line 25: @@
 Instead of assuming two's complement to allow for positive and negative exponents, IEEE 754 uses //biased// exponents: The exponent bits are interpreted as unsigned integer, but to get the "real" exponent value, you need to //substract// the bias value, which is ''FLOAT_MAX_EXP - 1'', ''DBL_MAX_EXP - 1'', or ''LDBL_MAX_EXP - 1'', respectively.
+=== Huh? ===
+Remember that IEEE 754 is a //floating point// standard. It makes //no// asumptions on the integer logic of the machine. What should the exponent be encoded at? Two's compliment? You don't know if the ALU supports that! So the exponent is stored unsigned. That means that the value ''1'' (1x10^0, or 1x2^0) is not stored with an exponent of all zeroes, but an exponent halfway between all zeroes (signifying denormals) and all ones (signifying INF / NaN).
 ==== Infinity ====
@@ Line 138: / Line 141: @@
 *: Signalling NaN (highest mantissa bit zero)
+The "Margin" is the difference between that number and its next higher "neighbor". Half that distance is where a decimal representation would be "tied" between those two binary representations.
+Take 0.25 and 0.3125 for example, which have a margin of 0.0625. Half that margin would be 0.03125. The decimal number (0.25 + 0.03125) = 0.28125 would be tied beween 0.25 and 0.3125. But 0.28124 would //unambiguously// identify the binary 0 001 00, because that's closer. This is how strtod() and scanf() can make use of this margin number.
+This works the other way around, too. Let's look at 0.4375. I don't need to //print// 0.4375 to unambiguously identify the binary 0 001 11, because either 0.43 or 0.44 would suffice (being less than 0.03125 away from the "real" value). Just 0.4 wouldn't do, because (0.4375 - 0.4) > 0.03125, and thus closer to 0.375 (0 001 10). This is the way printf() is looking at the issue.