Floating Point Modulo Operation

Question

I am trying to implement the range reduction operation for trigonometry. But instead I think it might be better to just perform a modulo pi/2 operation on incoming data. I was wondering what algorithms exist and are efficient for this operation for 32-bit IEEE 754 floating-point?

I have to implement this in assembly, so fmod, division, multiplication, etc. aren't available to me with just one instruction. My processor uses 16-bit words and I have implemented 32-bit floating point addition, subtraction, multiplication, division, square root, cosine, and sine. I just need range reduction (modulus) for inputting values to cosine and sine.

Answer 1

I think standard library's fmod() will be the best choice in most cases. Here's a link to a discussion of several simple algorithms.

On my machine, fmod() uses optimized inline assembly code (/usr/include/bits/mathinline.h):

#if defined __FAST_MATH__ && !__GNUC_PREREQ (3, 5)
__inline_mathcodeNP2 (fmod, __x, __y, \
  register long double __value;                           \
  __asm __volatile__                                  \
    ("1:    fprem\n\t"                            \
     "fnstsw    %%ax\n\t"                             \
     "sahf\n\t"                                   \
     "jp    1b"                               \
     : "=t" (__value) : "0" (__x), "u" (__y) : "ax", "cc");           \
  return __value)
#endif

So it actually uses a dedicated CPU instruction (fprem) for the calculation.

Answer 2

Maybe I'm missing the point here, but do you have anything against simply using fmod?

double theta = 10.4;
const double HALF_PI = 2 * atan(1);
double result = fmod(theta, HALF_PI);

Answer 3

The algorithm you want, to limit a floating point value between 0 and some modulus n:

Double fmod(Double value, Double modulus)
{
    return value - Trunc(value/modulus)*modulus;
}

for example pi mod e (3.14159265358979 mod 2.718281828459045)

3.14159265358979 / 2.718281828459045 
   = 1.1557273497909217179

Trunc(1.1557273497909217179)
   = 1

1.1557273497909217179 - 1
   = 0.1557273497909217179

0.1557273497909217179 * e
   = 0.1557273497909217179 * 2.718281828459045
   = 0.42331082513074800

pi mod e = 0.42331082513074800