Find centralized, trusted content and collaborate around the technologies you use most. Connect and share knowledge within a single location that is structured and easy to search. As you say, floats have about 7 digits of decimal precision, but the error in representation of both fixed-width decimals and floats is not uniformly logarithmic.
The relative error in rounding a number of the form 1. Similarly, depending on where a value falls in a binade, the relative error in rounding it to 24 binary digits can vary by a factor of nearly two. The upshot of this is that not all seven-digit decimals survive the round trip to float and back, but all six digit decimals do.
There are not that many six and seven digit decimals with exponents in a valid range for the float type; you can pretty easily write a program to exhaustively test all of them if you're still not convinced. It's really only 23 bits in the mantissa there's an implied 1, so it's effectively 24 bits, but the 1 obviously does not vary. This gives 6.
Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? On most machines today, floats are approximated using a binary fraction with the numerator using the first 53 bits starting with the most significant bit and with the denominator as a power of two. Many users are not aware of the approximation because of the way values are displayed. Python only prints a decimal approximation to the true decimal value of the binary approximation stored by the machine.
On most machines, if Python were to print the true decimal value of the binary approximation stored for 0. That is more digits than most people find useful, so Python keeps the number of digits manageable by displaying a rounded value instead.
Interestingly, there are many different decimal numbers that share the same nearest approximate binary fraction. For example, the numbers 0. Historically, the Python prompt and built-in repr function would choose the one with 17 significant digits, 0. Starting with Python 3. Note that this is in the very nature of binary floating-point: this is not a bug in Python, and it is not a bug in your code either. For more pleasant output, you may wish to use string formatting to produce a limited number of significant digits:.
One illusion may beget another. Between 10 -4 and 10 -3 there are five segments: [10 -4 , 2 , [2 , 2 , [2 , 2 , [2 , 2 , and [2 , 10 -3 ; they have decimal precision of 8, 7, 7, 7, and 6 digits, respectively. In any given segment, you can find examples of numbers that convert to a higher precision than supported by that segment:.
This is not precision, at least as we have defined it; precision is a property of a range of n-digit numbers, not a specific n-digit number. All seven of those numbers map to the same float, which in turn maps back to 9 digits to 1.
Broadly we can say that precision ranges from 6 to 8 digits — but can we say what the average precision is? There are powers of two in this range, most of which have a single precision value; those that cross a power of ten have two values. The denominator for my average was , the number of powers of two. For those powers of two with a single precision, I assigned a weight of 1.
For those powers of two split by powers of ten, I assigned a fractional weight, proportional to where the split occurs. With that methodology, I came up with an average decimal precision for single-precision floating-point: 7.
But it does tell you that you are likely to get 7 digits. If you care about the minimum precision you can get from a float, or equivalently, the maximum number of digits guaranteed to round-trip through a float, then 6 digits is your answer. If you could give only one answer, the safe answer is 6 digits. That way there will be no surprises print precision notwithstanding. Doing the same analysis for doubles I computed an average decimal precision of The three reasonable answers are 15 digits, digits, and slightly less than 16 digits on average.
The safe answer is 15 digits. Some will say 9 is the upper bound of precision for a float, and likewise, 17 digits is the upper bound for a double for example, see the Wikipedia articles on single-precision and double-precision. Longer mantissas would mean a more precise number, but smaller represantation because wxponent would have less bits. Shorter mantissas would mean the exact opposite, less precise numbers, but more space for exponent, so that higher or closest to zero number can be represented.
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This post aims to show you how to answer the questions: What precision do I have at a number? When will I hit precision issues? First, a very quick look at the floating point format. Floating Point Format Floating point numbers Wikipedia: IEEE have three components: Sign bit — whether the number is positive or negative Exponent bits — the magnitude of the number Mantissa bits — the fractional bits 32 bit floats use 1 bit for sign, 8 bits for exponent and 23 bits for mantissa.
What precision do I have at a number? A quick note on the maximum number you can store in floating point numbers, by looking at the half float specifically: A half float has a maximum exponent of 15, which you can see above puts the number range between and If you have a 30fps game, your frame delta is going to be 0. When will this happen though?
That means we need to ceil the number we got from the log2. With doubles, it falls apart at value ,,,, That is 8,, years!
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