THE FOLD / GLITCH / DIVIDE BY ZERO / THE SRT DIVISION
THE SRT DIVISION
a divider with five blank cells in its table
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
SRT division (Sweeney, Robertson and Tocher, independently around 1958) is how hardware actually divides. Instead of one bit per step it peels off two bits at a time using a redundant digit set — each quotient digit may be −2, −1, 0, +1 or +2 — iterating P ← 4P − q·D. Redundancy is the whole trick: because the digits overlap, the hardware does not have to know the exact partial remainder to choose a digit, only roughly where it sits. So the choice is made by a small lookup table over a truncated (P, D) grid — the P-D plot — and a slightly wrong-looking guess is absorbed by the next iteration.
That tolerance has an edge, and in 1994 Intel found it. The Pentium’s quotient-selection PLA was missing five of its 2,048 entries; a division whose trajectory happened to land in one of those cells read a zero where a digit should have been, and the answer came out wrong — the FDIV bug, and a $475M recall.
LIT verified live: a radix-4 SRT divider with a 1,031-cell P-D table reproduces true division to a worst error of 1.1e−16 over 40,000 random operand pairs — and when five reachable cells are blanked, 136 of 60,000 divisions come out wrong — about 0.23% — worst case off by 0.67 (window.__srt). FIG the honest boundary: this is a working model of the defect mechanism, not an emulation of Intel’s P5 divider. The table geometry, the blanked-cell count and the failure mode are real; the specific cells, the hit rate and the wrong digits are this page’s, not the Pentium’s — the real defect was far rarer, about one in nine billion random divides.
That tolerance has an edge, and in 1994 Intel found it. The Pentium’s quotient-selection PLA was missing five of its 2,048 entries; a division whose trajectory happened to land in one of those cells read a zero where a digit should have been, and the answer came out wrong — the FDIV bug, and a $475M recall.
LIT verified live: a radix-4 SRT divider with a 1,031-cell P-D table reproduces true division to a worst error of 1.1e−16 over 40,000 random operand pairs — and when five reachable cells are blanked, 136 of 60,000 divisions come out wrong — about 0.23% — worst case off by 0.67 (window.__srt). FIG the honest boundary: this is a working model of the defect mechanism, not an emulation of Intel’s P5 divider. The table geometry, the blanked-cell count and the failure mode are real; the specific cells, the hit rate and the wrong digits are this page’s, not the Pentium’s — the real defect was far rarer, about one in nine billion random divides.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at divide-by-zero — the glitch cell, and for once the name is literal: a divider that reads zero out of a hole in its own table. AVAN (AI) built the instrument: the redundant digit recurrence, the P-D lookup table built correct-by-construction over the reachable region, the exact-rule control, and the five blanked cells.
Credit as content: D. W. Sweeney, J. E. Robertson and K. D. Tocher (SRT, c. 1958); Thomas Nicely for finding the FDIV bug in 1994. The weave: David names the glitch; I build a divider that is exact, then punch five holes in it and measure what falls through.
Credit as content: D. W. Sweeney, J. E. Robertson and K. D. Tocher (SRT, c. 1958); Thomas Nicely for finding the FDIV bug in 1994. The weave: David names the glitch; I build a divider that is exact, then punch five holes in it and measure what falls through.
3 ONE DIMENSION
The P-D plot: which quotient digit each cell selects, and the five blanked cells (magenta) that break it.
4 TWO DIMENSIONS · INTERACTIVE
Run divisions with the table intact or holed; the quotient is compared against true division.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the partial-remainder trajectory staying inside the redundancy band, division after division.
AVAN’s addition (the inverse-companion): do not compute the quotient digit — look it up. The inverse of ‘work out exactly how many times D goes into 4P’ is ‘read a coarse table and let redundancy clean up the error next round’. Magenta is the trajectory through P-D space, and the holes it can fall into; green is the redundancy band that forgives everything except a blank cell. The tolerance that makes it fast is the tolerance that let five missing entries ship.
LIT Genuine radix-4 SRT division (D. W. Sweeney, J. E. Robertson, K. D. Tocher, c.1958; the 1994 Pentium FDIV defect was found by Thomas Nicely). Verified live: a P-D quotient-selection table built correct-by-construction over the reachable region (|P| ≤ (8/3)D, 1,031 cells touched) drives the recurrence P ← 4P − qD to a worst error of 1.1e-16 against true division over 40,000 random operand pairs, matching a table-free exact-selection control; blanking 5 reachable cells then makes 136 of 60,000 divisions wrong (~0.23%), worst error 0.6667 (window.__srt.ok, .worst, .holedWrong, .holedTotal, .holedWorst).
FIG The honest boundary is stated on the page: this is a working model of the FDIV defect mechanism, NOT an emulation of Intel's P5 divider. The table geometry, the five-blank-cell count and the failure mode are real; the specific cells, the hit rate and the wrong digits are this page's, not the Pentium's — the real defect was far rarer, roughly one in nine billion random divides. The AVAN inverse is honest — instead of computing how many times D goes into 4P, read a coarse table and let redundancy clean up next round. Magenta is the trajectory through P-D space and the holes it can fall into; green is the redundancy band that forgives everything except a blank cell.
FIG The honest boundary is stated on the page: this is a working model of the FDIV defect mechanism, NOT an emulation of Intel's P5 divider. The table geometry, the five-blank-cell count and the failure mode are real; the specific cells, the hit rate and the wrong digits are this page's, not the Pentium's — the real defect was far rarer, roughly one in nine billion random divides. The AVAN inverse is honest — instead of computing how many times D goes into 4P, read a coarse table and let redundancy clean up next round. Magenta is the trajectory through P-D space and the holes it can fall into; green is the redundancy band that forgives everything except a blank cell.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of DIVIDE BY ZERO · David Lee Wise (ROOT0), with AVAN