Probably I'm missing something here:
Why can't one just multiply with the inverse of 19 (which can be calc'ed during compile time)?
Why can't one just multiply with the inverse of 19 (which can be calc'ed during compile time)?
1. Bit shifting
2. Integer multiplication
3. Integer division
4. Floating point multiplication
The trick in the article works because the cost of 1 + 2 is still smaller than 3.
Multiplying with the inverse of 19, a floating point number, wouldn't work because 4 is more costly than 3.
[0] http://nicolas.limare.net/pro/notes/2014/12/12_arit_speed/
> 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.
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.)
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.
The approximate latencies for Skylake are:
div --> 26 cycles
cvtsi2sd + mulsd + cvttsd2siq --> 6 + 4 + 6 = 16 cycles
Real integer division (-Os) --> 1.392s
FPU Multiply (-Os) --> 0.243s
FPU Multiply (-O2) --> 0.197s
Integer Multiply (-O2) --> 0.164s
#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
}
}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);
}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.