/*
* Generate 624 more random values. This function is called when the
* state vector has been exhausted. It generates another batch of
* pseudo-random values. The performance of this function is critical
* to the performance of the Mersenne Twist PRNG, so it has been
* highly optimized.
*/
void mts_refresh(
register mt_state* state) /* State for the PRNG */
{
register int i; /* Index into the state */
register uint32_t*
state_ptr; /* Next place to get from state */
register uint32_t
value1; /* Scratch val picked up from state */
register uint32_t
value2; /* Scratch val picked up from state */
/*
* Start by making sure a random seed has been set. If not, set
* one.
*/
if (!state->initialized)
{
mts_seed32(state, DEFAULT_SEED32_OLD);
return; /* Seed32 calls us recursively */
}
/*
* Now generate the new pseudorandom values by applying the
* recurrence relation. We use two loops and a final
* 2-statement sequence so that we can handle the wraparound
* explicitly, rather than having to use the relatively slow
* modulus operator.
*
* In essence, the recurrence relation concatenates bits
* chosen from the current random value (last time around)
* with the immediately preceding one. Then it
* matrix-multiplies the concatenated bits with a value
* RECURRENCE_OFFSET away and a constant matrix. The matrix
* multiplication reduces to a shift and two XORs.
*
* Some comments on the optimizations are in order:
*
* Strictly speaking, none of the optimizations should be
* necessary. All could conceivably be done by a really good
* compiler. However, the compilers available to me aren't quite
* smart enough, so hand optimization needs to be done.
*
* Shawn Cokus was the first to achieve a major speedup. In the
* original code, the first value given to COMBINE_BITS (in my
* characterization) was re-fetched from the state array, rather
* than being carried in a scratch variable. Cokus noticed that
* the first argument to COMBINE_BITS could be saved in a register
* in the previous loop iteration, getting rid of the need for an
* expensive memory reference.
*
* Cokus also switched to using pointers to access the state
* array and broke the original loop into two so that he could
* avoid using the expensive modulus operator. Cokus used three
* pointers; Richard J. Wagner noticed that the offsets between
* the three were constant, so that they could be collapsed into a
* single pointer and constant-offset accesses. This is clearly
* faster on x86 architectures, and is the same cost on RISC
* machines. A secondary benefit is that Cokus' version was
* register-starved on the x86, while Wagner's version was not.
*
* I made several smaller improvements to these observations.
* First, I reversed the contents of the state vector. In the
* current version of the code, this change doesn't directly
* affect the performance of the refresh loop, but it has the nice
* side benefit that an all-zero state structure represents an
* uninitialized generator. It also slightly speeds up the
* random-number routines, since they can compare the state
* pointer against zero instead of against a constant (this makes
* the biggest difference on RISC machines).
*
* Second, I returned to Matsumoto and Nishimura's original
* technique of using a lookup table to decide whether to xor the
* constant vector A (MATRIX_A in this code) with the newly
* computed value. Cokus and Wagner had used the ?: operator,
* which requires a test and branch. Modern machines don't like
* branches, so the table lookup is faster.
*
* Third, in the Cokus and Wagner versions the loop ends with a
* statement similar to "value1 = value2", which is necessary to
* carry the fetched value into the next loop iteration. I
* recognized that if the loop were unrolled so that it generates
* two values per iteration, a bit of variable renaming would get
* rid of that assignment. A nice side effect is that the
* overhead of loop control becomes only half as large.
*
* It is possible to improve the code's performance somewhat
* further. In particular, since the second loop's loop count
* factors into 2*2*3*3*11, it could be unrolled yet further.
* That's easy to do, too: just change the "/ 2" into a division
* by whatever factor you choose, and then use cut-and-paste to
* duplicate the code in the body. To remove a few more cycles,
* fix the code to decrement state_ptr by the unrolling factor, and
* adjust the various offsets appropriately. However, the payoff
* will be small. At the moment, the x86 version of the loop is
* 25 instructions, of which 3 are involved in loop control
* (including the decrementing of state_ptr). Further unrolling by
* a factor of 2 would thus produce only about a 6% speedup.
*
* The logical extension of the unrolling
* approach would be to remove the loops and create 624
* appropriate copies of the body. However, I think that doing
* the latter is a bit excessive!
*
* I suspect that a superior optimization would be to simplify the
* mathematical operations involved in the recurrence relation.
* However, I have no idea whether such a simplification is
* feasible.
*/
state_ptr = &state->statevec[MT_STATE_SIZE - 1];
value1 = *state_ptr;
for (i = (MT_STATE_SIZE - RECURRENCE_OFFSET) / 2; --i >= 0; )
{
state_ptr -= 2;
value2 = state_ptr[1];
value1 = COMBINE_BITS(value1, value2);
state_ptr[2] =
MATRIX_MULTIPLY(state_ptr[-RECURRENCE_OFFSET + 2], value1);
value1 = state_ptr[0];
value2 = COMBINE_BITS(value2, value1);
state_ptr[1] =
MATRIX_MULTIPLY(state_ptr[-RECURRENCE_OFFSET + 1], value2);
}
value2 = *--state_ptr;
value1 = COMBINE_BITS(value1, value2);
state_ptr[1] =
MATRIX_MULTIPLY(state_ptr[-RECURRENCE_OFFSET + 1], value1);
for (i = (RECURRENCE_OFFSET - 1) / 2; --i >= 0; )
{
state_ptr -= 2;
value1 = state_ptr[1];
value2 = COMBINE_BITS(value2, value1);
state_ptr[2] =
MATRIX_MULTIPLY(state_ptr[MT_STATE_SIZE - RECURRENCE_OFFSET + 2],
value2);
value2 = state_ptr[0];
value1 = COMBINE_BITS(value1, value2);
state_ptr[1] =
MATRIX_MULTIPLY(state_ptr[MT_STATE_SIZE - RECURRENCE_OFFSET + 1],
value1);
}
/*
* The final entry in the table requires the "previous" value
* to be gotten from the other end of the state vector, so it
* must be handled specially.
*/
value1 = COMBINE_BITS(value2, state->statevec[MT_STATE_SIZE - 1]);
*state_ptr =
MATRIX_MULTIPLY(state_ptr[MT_STATE_SIZE - RECURRENCE_OFFSET], value1);
/*
* Now that refresh is complete, reset the state pointer to allow more
* pseudorandom values to be fetched from the state array.
*/
state->stateptr = MT_STATE_SIZE;
}
/**
Refresh the state for next set of random numbers.
*/
void mts_refresh(register mt_state* state /**< State for the PRNG */
){
register int i; /* Index into the state */
register mt_u32bit_t*
state_ptr; /* Next place to get from state */
register mt_u32bit_t
value1; /* Scratch val picked up from state */
register mt_u32bit_t
value2; /* Scratch val picked up from state */
/*
* Start by making sure a random seed has been set. If not, set
* one.
*/
if (!state->initialized)
{
mts_seed32(state, DEFAULT_SEED32_OLD);
return; /* Seed32 calls us recursively */
}
state_ptr = &state->statevec[MT_STATE_SIZE - 1];
value1 = *state_ptr;
for (i = (MT_STATE_SIZE - RECURRENCE_OFFSET) / 2; --i >= 0; )
{
state_ptr -= 2;
value2 = state_ptr[1];
value1 = COMBINE_BITS(value1, value2);
state_ptr[2] =
MATRIX_MULTIPLX(state_ptr[-RECURRENCE_OFFSET + 2], value1);
value1 = state_ptr[0];
value2 = COMBINE_BITS(value2, value1);
state_ptr[1] =
MATRIX_MULTIPLX(state_ptr[-RECURRENCE_OFFSET + 1], value2);
}
value2 = *--state_ptr;
value1 = COMBINE_BITS(value1, value2);
state_ptr[1] =
MATRIX_MULTIPLX(state_ptr[-RECURRENCE_OFFSET + 1], value1);
for (i = (RECURRENCE_OFFSET - 1) / 2; --i >= 0; )
{
state_ptr -= 2;
value1 = state_ptr[1];
value2 = COMBINE_BITS(value2, value1);
state_ptr[2] =
MATRIX_MULTIPLX(state_ptr[MT_STATE_SIZE - RECURRENCE_OFFSET + 2],
value2);
value2 = state_ptr[0];
value1 = COMBINE_BITS(value1, value2);
state_ptr[1] =
MATRIX_MULTIPLX(state_ptr[MT_STATE_SIZE - RECURRENCE_OFFSET + 1],
value1);
}
/*
* The final entry in the table requires the "previous" value
* to be gotten from the other end of the state vector, so it
* must be handled specially.
*/
value1 = COMBINE_BITS(value2, state->statevec[MT_STATE_SIZE - 1]);
*state_ptr =
MATRIX_MULTIPLX(state_ptr[MT_STATE_SIZE - RECURRENCE_OFFSET], value1);
/*
* Now that refresh is complete, reset the state pointer to allow more
* pseudorandom values to be fetched from the state array.
*/
state->stateptr = MT_STATE_SIZE;
}
