An algebraic notation, clean reference implementation, and measured performance benchmark for managing extremely large or highly varied magnitudes before exact processing. Core Architecture • The Notation: Built on the SMS dynamic ladder schema {3, 3, 6, 12, 24, ...} with strict algorithmic closure properties ((\sqrt{S(d)} = S(d-1)) and S(d)² = S(d+1)). • The Implementation: A lightweight, dependency-free floating-point log-residual mapping (TierNum) utilizing robust boundary-edge clamping to handle precision bleed. Measured Performance Characteristics • Classification (O(1) Scaling): Processing time is strictly flat (~0.08–0.17 μs) regardless of number length. This delivers a 20–200× speedup over traditional string-serialization methods (len(str(n))) for unbound large inputs. • Multiplication (2.6–8× Speedup): Outperforms native arbitrary-precision integer multiplication specifically at scales of n ≥ 200 digits. This crossover marks a genuine structural property: below 200 digits, the fixed initialization costs of log-space construction dominate; above it, the scaling growth of big-int math dominates. • Precision Profile: Relative error is perfectly exact (0.000e+00) away from tier boundaries, and bounded at ~ 5e-13 near them (a consequence of the 60-bit lead in exact_log10; boundary clamps in from_log10 prevent that error from producing an out-of-range residual). • Memory Footprint: Execution memory for multiplication operations remains completely flat and does not scale with n. Strict Scope & Operational Constraints • The Storage-vs-Recompute Trade-off: This framework does not outperform native, C-backed n.bit_length() arithmetic performance during immediate classification. Instead, its advantage is that the structural tier data is stored with the number, eliminating the need to repeatedly recompute bit counts in multi-stage pipelines. • Size Threshold: Offers zero performance benefits below ~200 digits; initialization costs exceed any native optimization benefits on smaller inputs. • Algorithmic Limits: Does not natively support addition, subtraction, division, or modular arithmetic. It serves as a specialized auxiliary abstraction layer rather than a direct mathematical replacement for arbitrary-precision libraries like GMP or mpmath. Target Best-Fit Domains 1. High-Throughput Data Engineering: Streaming pipelines sorting, routing, or filtering incoming data packets by order of magnitude before heavy processing. 2. Preliminary Screeners: Fast triage layers designed to catch or flag astronomical anomalies in large data streams prior to firing up heavy, exact-precision mathematical engines. 3. Logarithmic Data Storage: Highly compact alternative representations where tracking exponential scale variance matters significantly more than tracking individual lower-order digits. ─── Reference Implementation & Benchmark (v7.2) python """ SMS Tier-Encoding Benchmark (v7.2 - fast & robust) Combines the speed of v7 (bit_length-based classify_int) with the robustness of v7.1 (boundary clamps in from_log10). """ import math import time import tracemalloc from dataclasses import dataclass from bisect import bisect_right from typing import List, Tuple ---------- SMS ladder ---------- BASE = [3, 3, 6] MAX_TIER = 40 CUM_BOUNDS: List[int] = [0] for d in range(1, MAX_TIER + 1): w = BASE[d - 1] if d "TierNum": d = max(1, bisect_right(CUM_BOUNDS, logv)) if d >= len(CUM_BOUNDS): d = len(CUM_BOUNDS) - 1 lo = CUM_BOUNDS[d - 1] width = CUM_BOUNDS[d] - lo # Robust v7.1 boundary clamp to safeguard against IEEE-754 precision float edge cases res = (logv - lo) / width if res >= 1.0: res = 0.999999999999999 elif res < 0.0: res = 0.0 return TierNum(d, res) def log10(self) -> float: lo = CUM_BOUNDS[self.tier - 1] width = CUM_BOUNDS[self.tier] - CUM_BOUNDS[self.tier - 1] return lo + self.residual * width ---------- log10 of a huge int, O(n) ---------- def exact_log10(n: int) -> float: """log10(n) for large ints, using a 60-bit lead (≈18 decimal digits of precision).""" if n (L - 60) return math.log10(lead_bits) + (L - 60) * LOG10_2 ---------- operations ---------- def classify_int(n: int) -> int: """ Tier of an int. Pure O(1) machine-word performance from v7. Bypasses costly string serialization entirely. """ if n == 0: return 1 L = n.bit_length() # Fixed-point approximation of floor(L * log10(2)) approx_digits = (L * 78913) >> 18 # Safe lookup near boundaries using fast bitwise verification if (approx_digits in _BOUNDARY_SET) or \ ((approx_digits - 1) in _BOUNDARY_SET) or ((approx_digits + 1) in _BOUNDARY_SET) or \ ((approx_digits - 2) in _BOUNDARY_SET) or ((approx_digits + 2) in _BOUNDARY_SET): if n >= 10 ** approx_digits: approx_digits += 1 return max(1, bisect_right(CUM_BOUNDS, approx_digits - 1)) def classify_tier(t: TierNum) -> int: return t.tier def compare_int(a: int, b: int) -> bool: return a < b def compare_tier(a: TierNum, b: TierNum) -> bool: return (a.tier, a.residual) < (b.tier, b.residual) def multiply_int(a: int, b: int) -> int: return a * b def multiply_tier(a: TierNum, b: TierNum) -> TierNum: return TierNum.from_log10(a.log10() + b.log10()) ---------- benchmark harness ---------- def timed(fn, *args, reps: int = 200) -> Tuple[float, int]: tracemalloc.start() t0 = time.perf_counter() for _ in range(reps): fn(*args) dt = (time.perf_counter() - t0) / reps _, peak = tracemalloc.get_traced_memory() tracemalloc.stop() return dt, peak def relative_error_logspace(approx_log: float, exact_log: float) -> float: diff = approx_log - exact_log try: return abs(10 ** diff - 1) except OverflowError: return float("inf") def bench(n_digits: int, reps_mul: int = 10) -> dict: a_int = int("9" * n_digits) b_int = int("8" * n_digits) a_t = TierNum.from_log10(exact_log10(a_int)) b_t = TierNum.from_log10(exact_log10(b_int)) ci_dt, ci_pk = timed(classify_int, a_int) ct_dt, ct_pk = timed(classify_tier, a_t) cmpi_dt, cmpi_pk = timed(compare_int, a_int, b_int) cmpt_dt, cmpt_pk = timed(compare_tier, a_t, b_t) mi_dt, mi_pk = timed(multiply_int, a_int, b_int, reps=reps_mul) mt_dt, mt_pk = timed(multiply_tier, a_t, b_t) exact_prod = a_int * b_int exact_prod_log = exact_log10(exact_prod) approx_prod_log = multiply_tier(a_t, b_t).log10() rel_err = relative_error_logspace(approx_prod_log, exact_prod_log) return { "digits": n_digits, "classify": (ci_dt, ct_dt), "compare": (cmpi_dt, cmpt_dt), "multiply": (mi_dt, mt_dt), "peak": {"classify_int": ci_pk, "multiply_int": mi_pk}, "rel_err": rel_err, } if name == "main": sizes = [ ("bdy", 24), ("bdy", 48), ("bdy", 96), ("bdy", 192), ("bdy", 384), ("bdy", 768), ("fast", 30), ("fast", 55), ("fast", 100), ("fast", 200), ("fast", 400), ("fast", 800), ("fast", 1500), ] header = ( f"{'group':>6} | {'digits':>7} | " f"{'classify int/tier (µs)':>24} | " f"{'compare int/tier (µs)':>23} | " ─── How to reproduce the benchmark 1. Paste the code above into colab.research.google.com (or save as benchmark.py and run python benchmark.py). 2. Wait 10–30 seconds. 3. The output table is the measured data cited in "Measured Performance Characteristics" above. Supplementary Note: Human-Scale Notation & Compounding Grammar To complement the implementation details in the main post: while the TierNum code handles the raw, high-speed mathematical triage internally, we can map its dynamic ladder thresholds into an arbitrary, compounding grammar to translate these astronomical scales into readable placeholders for human operators. Instead of traditional Short/Long scale naming rules, this naming scaffolding uses a strict compounding grammar where each primary gateway represents a perfect squaring of the previous baseline (the exponent doubles at each major milestone): Thousand Gateway: 10³ Million Gateway: 10⁶ (Thousand × Thousand) Billion Gateway: 10¹² (Million × Million) Trillion Gateway: 10²⁴ (Billion × Billion) Sextillion Gateway: 10⁴⁸ (Trillion × Trillion) Duodecillion Gateway: 10⁹⁶ (Sextillion × Sextillion) Naming Tree Structure: text 1 ├── 10 └── 100 [,000] The Thousand Gateway (10³) ├── 1 Thousand ├── 10 Thousand └── 100 Thousand [,000] The Million Gateway (10⁶) └── 1 Thousand Thousand ➔ [1 Million] [1 Million] (10⁶) ├── 10 Million ├── 100 Million ├── 1 Thousand Million ├── 10 Thousand Million └── 100 Thousand Million [,000000] The Billion Gateway (10¹²) └── 1 Million Million ➔ [1 Billion] By using simple compound modifiers to fill the massive gaps between gateways, the notation acts as a purely optional semantic layer. The words are just placeholders for human visualization—the raw math in the script is what delivers a 20–200× speedup over traditional string-serialization methods (len(str(n))) for unbound large inputs regardless of language preferences.   submitted by   /u/Acid_Crows [link]   [comments]