TRANSCENDENTALS AS REFINEMENT INVARIANTS

x = 1.00
k =
DEFINITION
exp(x) := lim(k→∞) (1 + x/2^k)^(2^k)

Euler's method for y' = y:
• Step size: Δ = 2^(-k)
• Steps: N = 2^k
• y(1) = (1 + Δ)^N
CURRENT VALUES
Input: x = 1
Resolution: k = 10
Steps: 2^k = 1024
Estimate: 2.71828...
Actual: 2.71828...
INTEGER OPERATIONS
1. x >> k (divide by 2^k)
2. FP_ONE + shifted (1 + x/2^k)
3. Square k times (repeated squaring)
No floats needed!
x = 2.00
DEFINITION
log(x) := y such that exp(y) = x

Binary search with certified bounds:
• Maintain [lo, hi] where
• exp(lo) ≤ x ≤ exp(hi)
• Bisect until converged
BINARY SEARCH STATE
Input: x = 2
Lower: 0.693...
Upper: 0.693...
log(x): 0.693...
Actual: 0.693...
VERIFICATION
exp(log(x)) = 2.000...
Should equal x
θ = π/4
Iterations:
CORDIC ALGORITHM
(cos θ, sin θ) := CORDIC rotation of (1,0)

Each iteration (shifts + adds only):
• x' = x ∓ (y >> i)
• y' = y ± (x >> i)
• z' = z ∓ arctan(2^(-i))
CURRENT VALUES
θ = 0.785 rad
sin(θ) = 0.707...
cos(θ) = 0.707...
sin² + cos² = 1.000...
WHY CORDIC?
• NO multiplication needed!
• Only shifts (>> i) and adds (±)
• Each iteration adds 1 bit precision
Perfect for hardware / bitwise
θ from 0 to π: π
EULER'S IDENTITY
e^(iπ) = -1

Or equivalently:
e^(iπ) + 1 = 0
AS REFINEMENT INVARIANTS
exp(iθ) := (cos θ, sin θ)

At θ = π (half-turn):
cos(π) = -1
sin(π) = 0

Therefore:
exp(iπ) = (-1, 0) = -1
CURRENT ROTATION
θ = π
exp(iθ) = (-1, 0)
|exp(iθ)| = 1.000
THE COINCIDENCE
Three invariants meet:
• e (growth invariant)
• π (area/rotation invariant)
• i (rotation by 90°)

All defined from integers!