Fast pseudorandom-number generators with modulus 2 <sup>k</sup> or 2 <sup>k</sup> -1 using fused multiply-add
Bibliographic record
Abstract
Many numerically intensive computations done in a scientific computing environment require uniformly distributed pseudorandom numbers in the range (0, 1) and (−1, 1). For multiplicative congruential generators with modulus 2k, k ≤ 52, and period 2k-2, we show that the cost per random number for these two distributions is 3 and 3.125 multiply–adds on RS/6000® processors. Our code, on the IBM POWER2 Model 590, produces more than 40 million uniformly distributed pseudorandom numbers per second for both ranges (0, 1) and (−1, 1). Additionally, our code sustains the 40 million per second rate for data out of cache. The Numerical Aerodynamic Simulation (NAS) parallel benchmarks use a linear congruential generator with modulus 246. Our result is about 50 times faster than the generic implementation given in the benchmarks. The extra-accuracy fused multiply-add instruction of RS/6000 machines combined with a few algorithmic innovations gives rise to the 50-fold increase. If IEEE 64-bit arithmetic is used with our Fortran code on POWER and PowerPC® architectures, the results we obtain are bit-wise identical to the generic algorithms. The paper gives several illustrations of a general technique called the Algorithm and Architecture approach. We demonstrate herein that programmer-controlled unrolling of loops is equivalent to “customized vectorization of RISC-type code.” Customized vectorization is more powerful than ordinary vectorization, and it is only possible on RISC-type machines. We illustrate its use to show that RS/6000 processors can compute the distribution (−1, 1) at the rate of 3.125 multiply–adds. We also specify a linear congruential generator that is related to the multiplicative congruential generator referred to above. It has a full period of 2k, where 2kis the modulus. The cost per random number [in the range (0, 1)] for this generator is four multiply–adds on RS/6000 processors. Our code, on the IBM POWER2 Model 590, for this generator produces more than 30 million uniformly distributed pseudorandom numbers per second for the range (0, 1). We show that this generator is “embarrassingly parallel,” or EP. Using the Algorithm and Architecture approach, we describe a new concept called “generalized unrolling.” Finally, we present a multiplicative congruential generator for which the modulus is not a power of 2. Such a generator, as well as one with modulus 2k, is selectable as the generator used in the RANDOM_NUMBER intrinsic function of IBM XL Fortran and XL High Performance Fortran. All of the generators reported here are EP. Using an IBM SP2 machine with 250 wide nodes, it is possible to compute more than ten billion uniform random numbers in a second.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.011 | 0.007 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".