Floating Point Representation

Goal of this exercise is to better understand the IEEE 754 representation of floating points, the behavior of operations among them and fp-exceptions.

run paranoia: is your system IEEE 754 conformant?
Using the "float-as-int union" print exponent and mantissa of a given number, find "next and previous float", verify the representation of infinites, subnormals and NaN.

suggest code to identify subnormals, NaN and infinites either checking their binary representation or using their "properties"

Write simple operations among fp and verify the results w.r.t. the expectations of the equivalent expression using "real numbers".

measure the ulp-gap (and its size as fp-number!) between expectations and results

Make them generating exceptions and verity which exceptions have been actually raised.
Trap the exceptions using SIGFPE

improve the accuracy of operations using refactoring, Kahan summation or other technique.

compare the speed w.r.t. using double-precision

Apply all this to the minimization exemple

Code

in exercises:
lookInFloat.cpp fpe.cpp kernels.cc loglike.cpp
in examples:
paranoia.c

Hints

export CXX=g++-451
export CXXFLAGS="-O2 -std=gnu++0x -msse4 -ftree-vectorize -ftree-vectorizer-verbose=1 -pthread -fPIC -fopenmp"
export LD_LIBRARY_PATH=${PWD}/lib:${LD_LIBRARY_PATH}

// use extern to avoid inlining (compile with -fPIC)
// "volatile" can be used to avoid eager optimization


objdump -S -r -C --no-show-raw-insn -w test.o | less
(on MacOS: otool -t -v  -V -X test.o  | c++filt | less)

References

IEEE 754 on Wikipedia
Single precision floating point format
man fetestexcept
std::numerical_limits