FEFCS Calculator
Fractional Exact Flow Computing System — Al-Khwarizmi Cycle
Interactive demonstration of the mathematical innovation by Miguel Vidal Bravo-Jandia:
exact fractions vs decimal approximations — the systemic bias revealed.
What is FEFCS?
The FEFCS system (Fractional Exact Flow Computing System) is a calculation method based on exact fractional arithmetic, rooted in the work of Al-Khwarizmi. It reveals and corrects the systematic rounding errors of the standard decimal system — errors that are invisible but financially significant at scale.
The problem
The standard decimal system introduces rounding errors at every non-integer division. Multiplied across millions of banking transactions, these approximations generate systematic overcharges at the debtor's expense.
The FEFCS solution
FEFCS works exclusively with exact fractions and pure addition — no multiplication, no rounding. Every 3-iteration cycle, the result lands on an exact integer (universal gateway X 0/0 = X.00).
Who is it for?
Researchers, finance professionals, developers and curious minds seeking to understand the limits of decimal computation and explore a mathematically rigorous alternative, applicable to banking, insurance and cryptography.
How to use the calculator
Choose a scenario or enter your values
Click one of the sector scenarios (Banking, Crypto, Trading…) to automatically load real parameters, or directly enter your numerator, denominator and number of iterations.
Read the results side by side
The ✅ FEFCS exact panel shows the precise fractional result. The ⚠️ Decimal panel shows the approximations accumulated by the standard IEEE 754 system.
Observe the financial overcharge
The Financial impact block automatically calculates the overcharge generated by the decimal system — per transaction, per thousand, per million and per billion operations.
Test the universal gateway
Try the 🎯 Key gateway scenario (20 ÷ 3, 3 iterations): FEFCS reaches 7 0/0 = 7.00€ exact while the decimal accumulates errors — this is the core demonstration of the innovation.
🎛️ Calculation parameters
🎯 Sector scenarios
Click a scenario to automatically load the parameters
🔒 Al-Khwarizmi FEFCS System
💥 Standard Decimal System
💰 Debtor overcharge (decimal system)
🔬 Comparative analysis
📚 Methodology — Founding principles
🎯 Al-Khwarizmi Cycle
Base: Euclidean division (a ÷ b = q remainder r)
Rule: Each iteration = pure addition of r/b
Reset: Iterations 3, 6, 9… → X 0/0 (exact integer)
Gateway: X 0/0 = X.00 in decimal
⚖️ Miguel Convention
Addition: a/b + c/d = (a+c)/(b+d)
Forbidden: No multiplication (pure Al-jabr)
Principle: Exact fractions that land precisely
Result: Zero rounding error
💰 Financial impact
EU regulation: Systemic banking rounding
Effect: 1 in 10 transactions unfavorable to debtor
Beneficiary: The creditor (bank/lender)
Solution: FEFCS eliminates this injustice
🔍 Exact sequence (20 ÷ 3)
- 0 iter. → 6⅔
- 1 iter. → 6 4/6
- 2 iter. → 6 6/9
- 3 iter. → 7 0/0 = 7.00€ ✅
- 4 iter. → 7⅔
- 6 iter. → 8 0/0 = 8.00€ ✅