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Zaneham merged 1 commit into from
Jan 27, 2026
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fixing documentation #5

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48 changes: 12 additions & 36 deletions DEVIATIONS.md
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Expand Up @@ -105,45 +105,21 @@ The following functions produced **different** results with -O3 -ffast-math:

**Analysis**: 4 ULPs difference in DP. Hex differs by 4 in last digit (0x54 vs 0x58).

## CRITICAL: Algorithmic Accuracy Failures
## Large Argument Bessel Functions - Verified Correct

The following are **NOT** optimisation-dependent - these are fundamental accuracy
failures in SLATEC's large-argument and high-order handling. These failures occur
identically across all compiler configurations (-O0, -O2, -O3 -ffast-math).
SLATEC's Bessel function implementations have been verified against mpmath (arbitrary
precision arithmetic, 100+ digits) for large arguments:

### Bessel J - Large Argument Failures
| Function | SLATEC Computed | mpmath (100 digit) | Status |
|----------|-----------------|---------------------|--------|
| J_0(500) | -0.0341005568807320 | -0.0341005568807320 | ✓ Correct |
| J_0(1000) | +0.0247866861524202 | +0.0247866861524202 | ✓ Correct |
| J_9(20) | 0.1251262546479942 | 0.1251262546479942 | ✓ Correct |
| Y_0(50) | -0.0980649954700771 | -0.0980649954700771 | ✓ Correct |
| Y_0(100) | -0.0772443133650832 | -0.0772443133650832 | ✓ Correct |

| Function | NIST Reference | SLATEC Computed | Error Type |
|----------|----------------|-----------------|------------|
| J_0(500) | +0.0179753822938868 | **-0.0341005568807317** | **WRONG SIGN, 290% ERROR** |
| J_0(1000) | -0.0246711376936846 | **+0.0247866861524200** | **WRONG SIGN** |

### Bessel J - High Order Failure

| Function | NIST Reference | SLATEC Computed | Error Type |
|----------|----------------|-----------------|------------|
| J_9(20) | 0.2453202954891665 | **0.0** | **TOTAL LOSS OF ACCURACY** |

### Bessel Y - Large Argument Failures

| Function | NIST Reference | SLATEC Computed | Error Type |
|----------|----------------|-----------------|------------|
| Y_0(50) | -0.0560404718523358 | **-0.0980649954700770** | **75% ERROR** |
| Y_0(100) | -0.0772433752531550 | -0.0772443133650831 | 0.01% error |

### Root Cause Analysis

These failures likely stem from:
1. **Range reduction errors** in trigonometric calculations for large arguments
2. **Recurrence instability** for high-order Bessel functions
3. **Asymptotic expansion limitations** not properly bounded

### Recommendations for Critical Applications

1. **DO NOT USE** J_n(x) for x > 100 without independent verification
2. **DO NOT USE** Y_n(x) for x > 20 without independent verification
3. **DO NOT USE** J_n(x) for n > 5 and x > 15 without verification
4. For large arguments, consider alternative implementations (e.g., AMOS library)
**Note:** An earlier version of this document contained incorrect reference values
that were mistakenly attributed to NIST. The SLATEC implementations are accurate.

---

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