> For the complete documentation index, see [llms.txt](https://docs.moduluszk.io/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.moduluszk.io/performance-the-prover-envelope/like-for-like-fib-10-000-same-algorithm-everywhere.md).

# Like for like: fib(10,000), same algorithm everywhere

Every system proves fib(10,000) mod 7919 by fast doubling, fourteen rounds instead of ten thousand steps. The comparator guests use a hand-written doubling routine; zkFOL is handed the naive two-call recurrence and the compiler rewrites it to the doubling form. A second zkFOL row proves the exact 2,090-digit integer with no modular reduction: the harder claim, included because the reduction is not what buys the result. 

| Workload: fib(10,000) mod 7919, fast doubling  | Prove    | Verify  | Prover memory |
| ---------------------------------------------- | -------- | ------- | ------------- |
| zkFOL, compiler-derived doubling               | 11.48 ms | 2.30 ms | under 10 MB   |
| zkFOL, exact 2,090-digit integer, no reduction | 9.98 ms  | 3.88 ms | under 10 MB   |
| RISC Zero, composite                           | 3.69 s   | 11.8 ms | 312 MB        |
| RISC Zero, succinct                            | 14.73 s  | 12.4 ms | 1.39 GB       |
| SP1, core                                      | 13.32 s  | 74.3 ms | 9.39 GB       |
| SP1, compressed                                | 49.24 s  | 32.9 ms | 17.00 GB      |

Prove-time ratios against the reduced zkFOL row: 321x, 1,280x, 1,160x and 4,290x for the four comparator rows in table order; against the unreduced row, which computes the whole integer while the guests compute a four-digit remainder, 370x to 4,930x. Memory ratios run from roughly 30x to roughly 1,700x. \[S, benchmark record]