> IMPORTANT: Useful feedback revealed that some of these measures are seriously flawed. A major update is on the way.
Looking over the results, some of the numbers are off.
On Intel CPUs, FP multiplication is faster than integer division. Might not be true on ARM CPUs which generally have slower FPUs.
On Skylake, for example, 32-bit unsigned integer division has a 26 cycle latency with a throughput of 1 instruction / 6 cycles, while 32/64-bit floating point multiplication has a 4 cycle latency with a throughput of 2 instructions / cycle.
The crucial point, however, is that while FP multiplication is faster than integer division, converting between a floating point and an integer is very slow: cvtsi2ss and cvtss2si have a latency of 6 cycles each. This adds a latency of 12 cycles for each of these multiplications.
For divisions by a constant value that don't easily decompose into shifts you can fall back to multiplication by a magic constant which is the integer reciprocal. (This is also something compilers do and is what's being explained in the article.)
Multiplying by the integer reciprocal only works if the dividend is an integer multiple of the divisor.
What's being explained in the article is multiplying by a fraction the value of which is close to the rational reciprocal of the divisor, and where the denominator of the fraction is an integer power of two (so dividing by the denominator can be done with a shift).
The fraction in this case is (2938661835 + 2^32) / 2^37.
#include <stdio.h>
int main() {
volatile unsigned x;
for (unsigned n = 0; n < 100000000; ++n) {
#if 1 /* Change to 0 to use FPU. */
/*
Compile with -Os to get GCC to emit div instruction.
-O2 to emit integer multiply.
Clang emits integer multiply, even with -Os.
*/
x = n / 19;
#else
/* Use the FPU. */
x = (double)n * (1.0 / 19.0);
#endif
}
}
It works for 32-bit unsigned integers and double precision floats.
For n < 19, "(double)n * (1.0 / 19.0)" evaluates to a double between 0.0 and 1.0, then it is truncated to 0 when it is implicitly converted to unsigned int.
Since there are only 2^32 values for 32-bit integers, it is possible to test all values in under a minute:
#include <stdio.h>
#include <stdint.h>
int main() {
uint32_t n = 0;
do {
uint32_t a = n / 19;
uint32_t b = (double)n * (1.0 / 19.0);
if (a != b) {
printf("Not equal for n = %u\n", n);
}
++n;
} while (n != 0);
}
Note that approach done naively will produce wrong answers for lots of divisors, because the reciprocal will not be exactly representable. For example, 49*(1.0/49) is less than one.
If you round the FP division up (to the next largest representable value, that is), it should be correct, for 32 bit integer types at least.
> On Skylake [...] 32/64-bit floating point multiplication has a 4 cycle latency with a throughput of 2 instructions / cycle.
Of course, there are some operations that are very expensive (trigonometric functions, for example), but they're not necessary here, and they're also very expensive on the GPU.
Why can't one just multiply with the inverse of 19 (which can be calc'ed during compile time)?