In the context of argument reduction for trigonometric functions, what you are looking at is Cody-Waite argument reduction, a technique introduced in the book: William J. Cody and William Waite, Software Manual for the Elementary Functions, Prentice-Hall, 1980. The goal is to achieve, for arguments up to a certain magnitude, an accurate reduced argument, despite subtractive cancellation in intermediate computation. For this purpose, the relevant constant is represented with more than native precision, by using a sum of multiple numbers of decreasing magnitude (here: DP1, DP2, DP3), such that all of the intermediate products except the least significant one can be computed without rounding error.
Consider as an example the computation of sin (113) in IEEE-754 binary32 (single precision). The typical argument reduction would conceptually compute i=rintf(x/(π/2)); reduced_x = x-i*(π/2). The binary32 number closest to π/2 is 0x1.921fb6p+0. We compute i=72, the product rounds to 0x1.c463acp+6, which is close to the argument x=0x1.c40000p+6. During subtraction, some leading bits cancel, and we wind up with reduced_x = -0x1.8eb000p-4. Note the trailing zeros introduced by renormalization. These zero bits carry no useful information. Applying an accurate approximation to the reduced argument, sin(x) = -0x1.8e0eeap-4, whereas the true result is -0x1.8e0e9d39...p-4. We wind up with large relative error and large ulp error.
We can remedy this by using a two-step Cody-Waite argument reduction. For example, we could use pio2_hi = 0x1.921f00p+0, and pio2_lo = 0x1.6a8886p-17. Note the eight trailing zero bits in single-precision representation ofpio2_hi, which allow us to multiply with any 8-bit integer i and still have the product i * pio2_hi representable exactly as a single-precision number. When we compute ((x - i * pio2_hi) - i * pio2_lo), we get reduced_x = -0x1.8eafb4p-4, and therefore sin(x) = -0x1.8e0e9ep-4, a quite accurate result.
The best way to split the constant into a sum will depend on the magnitude of i we need to handle, on the maximum number of bits subject to subtractive cancellation for a given argument range (based on how close integer multiples of π/2 can get to integers), and performance considerations. Typical real-life use cases involve two- to four-stage Cody-Waite reduction schemes. The availability of fused multiple-add (FMA) allows the use of constituent constants with fewer trailing zero bits. See this paper: Sylvie Boldo, Marc Daumas, and Ren-Cang Li, "Formally verified argument reduction with a fused multiply-add." IEEE Transactions on Computers, 58 :1139–1145, 2009. For a worked example using fmaf() you might want to look at the code in one of my previous answers.
