CORDIC Hyperbolic IP

Rotation Mode – Parameter Sensitivity & Accuracy Characterization

1. Configuration Under Test

2. Error Metrics

• Max Error: Maximum absolute deviation across input sweep
• RMS Error: Root-mean-square error across input sweep
• Reference values computed using double-precision sinh(), cosh(), tanh()

3. Iteration Count Sweep (Convergence Depth)

Observations

• Sinh/Cosh error decreases exponentially with iteration depth
• Beyond ~ITER = 16, improvement is limited by fixed-point quantization
• Tanh error does not improve with additional iterations
• Tanh error is dominated by implementation structure (division / saturation behavior)

4. Fraction Bits Sweep (Phase Quantization)

Observations

• Sinh/Cosh error scales approximately with 2^(-FRAC_BITS)
• Below ~24 bits, quantization dominates
• Tanh accuracy is largely unaffected by additional fractional precision
• Error floor for Tanh is structural, not precision-limited

5. Internal Width Sweep (Datapath Precision)

Observations

• WIDTH < 28 introduces noticeable amplitude distortion
• WIDTH ≥ 32 provides stable convergence
• Excessive width with mismatched scaling causes catastrophic numerical failure
• Padding/pruning warnings correlate with instability

6. Output Width & Scaling

6.1 Correct Scaling (Reference)

6.2 Reduced Output Width (Quantization Noise)

6.3 Incorrect Scaling (Failure Region)

Observations

• Correct output scaling is mandatory
• Underscaling causes amplitude collapse
• Overscaling causes sign inversion and catastrophic distortion
• RMS error saturates near ~0.7–1.2 in failure regions

7. Tanh Accuracy Summary

Observations

• Tanh error is dominated by structural implementation behavior
• Large absolute deviation near ±1 persists
• Tanh should be treated as best-effort output

8. Key Takeaways (Rotation Mode)

• ITER = 16 sufficient for convergence
• FRAC_BITS ≥ 28 recommended
• WIDTH ≥ 32 ideal
• Correct output scaling is mandatory
• Tanh accuracy is structurally limited

CORDIC Hyperbolic IP

Vectoring Mode – Parameter Sensitivity & Accuracy Characterization

1. Configuration Under Test

Shift schedule (with required repeats): [1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 13, 14]

2. Error Metrics

• Max Error: Maximum absolute deviation
• RMS Error: Root-mean-square deviation
• Reference values computed using double-precision exp() and ln()

3. EXP Accuracy (Rotation Mode Result — Listed Here for Comparison)

Note: exp is implemented under Hyperbolic Rotation Mode. It is shown here only for comparison with LN, which uses vectoring.

Configuration

• OUT_SHIFT = 17
• Input range: −1.0 to +1.0

Results

Observations

• Stable and monotonic across full ±1.0 range
• No overflow observed within test domain
• Error consistent with fixed-point quantization limits
• Correct repeat scheduling resolves prior instability

EXP rotation mode is now numerically stable in tested range.

4. LN Accuracy (Vectoring Mode)

Configuration

• OUT_SHIFT = 16
• Input range: 0.2 to 1.9

Results

Observations

• Converges correctly across full tested domain
• No collapse or constant-output behavior
• Symmetric error distribution around reference
• Residual error dominated by fixed-point truncation
• Proper shift schedule resolves prior convergence defect

LN vectoring mode is now numerically stable in tested range.

5. Key Takeaways (Vectoring Mode)

• Hyperbolic repeat scheduling is mandatory for stability
• ITER = 16 sufficient for convergence at 30 fractional bits
• WIDTH = 32 provides stable internal dynamic range
• EXP (rotation mode) and LN (vectoring mode) both achieve ~1e−4 RMS accuracy
• No structural instability observed within validated ranges
• Current implementation is production-ready within defined domain