Explore direct neighbors and limits of IEEE floating-point values. This program can be used as educational tool to delve deep into float rounding errors, epsilons and explore the properties and limits of IEEE floats.
- Precision: 16-bit, 32-bit, 64-bit, 128-bit*
- Number of neighbors
- Endianness: little-endian, big-endian, machine-default
*: machine-dependent (usually padded or unavailable)
python float_neighbors.py 0x0000 -p 16 -e "<" -n 2
Offset | Hex value | Numerical value
+2 | 0x0200 | 1e-07
+1 | 0x0100 | 6e-08
+0 | 0x0000 | 0.0
-1 | 0x0180 | -6e-08
-2 | 0x0280 | -1e-07
python float_neighbors.py " -inf" -p 32 -n 2
Offset | Hex value | Numerical value
+2 | 0xff7ffffe | -3.4028233e+38
+1 | 0xff7fffff | -3.4028235e+38
+0 | 0xff800000 | -inf
python float_neighbors.py 2 -p 16 -n 2
Offset | Hex value | Numerical value
+2 | 0x4002 | 2.004
+1 | 0x4001 | 2.002
+0 | 0x4000 | 2.0
-1 | 0x3fff | 1.999
-2 | 0x3ffe | 1.998
Python floats are internally 64 bits in size and use machine-default endianness. By only using NumPy arrays for data, we circumvent any internal Python conversions. Furthermore, we are now able to leverage custom NumPy datatypes.
A custom NumPy float datatype could for example be np.dtype('>f4') where > indicates big-endianness, f indicates float storage and 4 indicates that 2 bytes are used for storage.
Obtaining the next closest numerical float value is not trivial. Fortunately, np.nextafter/2 is implemented in the NumPy specification, which returns the direct next possible float neighbor (towards np.NINF and np.PINF) and handles edge cases when passing through any float category:
inf > normal > subnormal > 0.0 > -subnormal > -normal > -inf
The infinity values used for retrieving neighbors have to be treated with special care: they are put in an array with the custom dtype and subsequently retrieved to maintain precision and byte order.
Runtime complexity of the program is O(n) as the previous float is always used to calculate the next float.
Memory complexity of the program is O(1) as the program initially walks up and subsequently walks down to calculate and display all the floats one by one in descending order.
Converting a float into its hex representation is not trivial. Luckily, NumPy provides a method to convert an entire NumPy array to a bytes object: tobytes/0. We use this method on a NumPy array containing a single float and then use Python's hex/0 to convert it into hex format.
Converting a hex representation into a float is done in two steps. We can simply use bytes.fromhex/1 to obtain the bytes representation and then rebuild the (1-element) float array using np.frombuffer/2 with our custom dtype.
Printing a 128-bit float directly within an f-string is not possible: internally Python will convert the number to a 64-bit float. Hence the print statements for the numerical values are on separate lines: this will circumvent conversion and properly print the float to its full precision.
This was an educational project to combine the flexibility and elegance of Python with the low-level power of C provided by the NumPy interface. The main challenge was to maintain precision and endianness of the stored data while performing intermediate operations.