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    <p>To understand de-normal floating point values you first have to understand normal ones. A floating point value has a mantissa and an exponent. In a decimal value, like 1.2345E6, 1.2345 is the mantissa, 6 is the exponent. A nice thing about floating point notation is that you can always write it normalized. Like 0.012345E8 and 0.12345E7 is the same value as 1.2345E6. Or in other words, you can always make the first digit of the mantissa a non-zero number, as long as the value is not zero.</p> <p>Computers store floating point values in binary, the digits are 0 or 1. So a property of a binary floating point value that is not zero is that it can always be written starting with a 1.</p> <p>This is a very attractive optimization target. Since the value always starts with 1, <em>there is no point in storing that 1</em>. What is nice about it is that you in effect get an extra bit of precision for free. On a 64-bit double, the mantissa has 52 bits of storage. The actual precision is 53 bits thanks to the implied 1.</p> <p>We have to talk about the smallest possible floating point value that you can store this way. Doing it in decimal first, if you had a decimal processor with 5 digits of storage in the mantissa and 2 in the exponent then the smallest value it could store that isn't zero is 1.00000E-99. With 1 being the implied digit that isn't stored (doesn't work in decimal but bear with me). So the mantissa stores 00000 and the exponent stores -99. You cannot store a smaller number, the exponent is maxed-out at -99.</p> <p>Well, you can. You could give up on the normalized representation and forget about the implied digit optimization. You can store it <em>de-normalized</em>. Now you can store 0.1000E-99, or 1.000E-100. All the way down to 0.0001E-99 or 1E-103, the absolute smallest number you can now store.</p> <p>This is in general desirable, it extends the range of values you can store. Which tends to matter in practical computations, very small numbers are very common in real-world problems like differential analysis.</p> <p>There's however also a big problem with it, you lose accuracy with de-normalized numbers. The accuracy of floating point calculations is limited by the number of digits you can store. It is intuitive with the fake decimal processor I used as an example, it can only ever compute with 5 significant digits. As long as the value is normalized, you always get 5 significant digits.</p> <p>But you'll lose digits when you de-normalize. Any value between 0.1000E-99 and 0.9999E-99 has only 4 significant digits. Any value between 0.0100E-99 and 0.0999E-99 has only 3 significant digits. All the way down to 0.0001E-99 and 0.0009E-99, only one significant digit left.</p> <p>This can greatly reduce the accuracy of the final calculation result. What's worse, it does so in a highly unpredictable manner since these very small de-normalized values tend to show up in a more involved calculation. That's certainly something to worry about, you cannot really trust the end result anymore when it has only 1 significant digit left.</p> <p>Floating point processors have ways to let you know about this or otherwise sail around the problem. They can for example generate an interrupt or signal when a value becomes de-normalized, letting you interrupt the calculation. And they have a "flush-to-zero" option, a bit in the status word that tells the processor to automatically convert all de-normal values to zero. Which tends to generate infinities, an outcome that tells you that the result is junk and should be discarded.</p>
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    1. POWhat is this "denormal data" about ? - C++
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    1. COquestion: who makes this choices ? As programmer i can declare, assign and use float values, but who manages this decisions about implementation details ? the hardware or the software ( compiler i guess ) ? And based on what ?
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    2. COThe implementation details were picked by the chip designer. The way the floating point processor is programmed to deal with de-normals is up to the programmer. Whether or not that's important is up to the algorithm designer that knows the domain.
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    3. COcan you make an example about an algorithm that cares about this ?
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