You're probably more of a "real programmer" than whomever wrote this code--it's numerics code, after all.
There's no real tricks or anything to this code, it's just a polynomial evaluation of a Taylor series (which you probably learned, and subsequently forgot, from calculus), with some steps done using techniques for getting higher precision out of a floating-point value, and (in some other functions in libm, not here) occasional use of the bit representation of floating-point values. It's standard techniques, but in a field that is incredibly niche and where there is very little value in delving into if you don't have a need for it.
There's no real tricks or anything to this code, it's just a polynomial evaluation of a Taylor series (which you probably learned, and subsequently forgot, from calculus), with some steps done using techniques for getting higher precision out of a floating-point value, and (in some other functions in libm, not here) occasional use of the bit representation of floating-point values. It's standard techniques, but in a field that is incredibly niche and where there is very little value in delving into if you don't have a need for it.