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Here are a few minor suggestions for squeezing out a little extra speed in your function.

In the Cython code you wrote above you use the Python abs and int function. This induces coercion of your fast C-data types to slow Python-data types. I suggest starting your speedups by using the C-abs function and <int>, instead.

%cython

cdef extern from "math.h":
    double fabs(double)

cpdef double g2(double x):
    x = fabs(x)
    x = 4.0*(x/4.0 - <int>(x/4.0))
    if x <= 2.0:
        return 1.0-x
    return -3.0+x

I tried using floor(fabs(x/4.0)) instead of <int>(x/4.0) but I didn't get any additional speedup. Also, cpdef improves performance if you call this function from Cython code.

Here are a few minor suggestions for squeezing out a little extra speed in your function.

In the Cython code you wrote above you use the Python abs and int function. This induces coercion of your fast C-data types to slow Python-data types. I suggest starting your speedups by using the C-abs function and <int>, instead.

%cython

cdef extern from "math.h":
    double fabs(double)

cpdef double g2(double x):
    x = fabs(x)
    x = 4.0*(x/4.0 - <int>(x/4.0))
    if x <= 2.0:
        return 1.0-x
    return -3.0+x

I tried using floor(fabs(x/4.0)) instead of <int>(x/4.0) but I didn't get any additional speedup. Also, cpdef improves performance if you call this function from Cython code.code. In particular, the $f(x)$ you define above should be a Cython function as well so you can take full advantage of the Cython speed benefits.