On accelerated methods to evaluate sums of products of rational numbers
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Bibliographic record
Abstract
In this paper we consider the problem of fast computation of sums of n-ary products of rational numbers, for large n. We present improvements to the standard binary splitting algorithm which are due to numerous factors, including changing the standard arbitrary precision integer representation to one that is more suitable for such computations, unrolling, and chains of recurrences techniques. For the computation of ζ(3) to 640000 decimal digits, we achieve a speedup factor of 2.65 over the standard binary splitting algorithm, which compares favorably to the ideal case in which the numerator and the denominator can be reduced by their greatest common divisor at no cost. If asymptotically fast multiplication is not available (as in the Java Development Kit), a speedup of an order of magnitude is easily obtained.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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 it