Source: https://guthlabs.ai/docs/instruments/ieee-754-float-inspector/mechanism/

instrument mechanism / verified-clean

# IEEE-754 Float Inspector mechanism

The inspectable method record for IEEE-754 Float Inspector, kept separate from implementation and qualification.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## Model scope

The current honest label is appropriate but incomplete. The instrument should explicitly state that the displayed reconstruction formula applies only to normalized finite values, define all IEEE-754 special cases, specify decimal conversion and rounding behavior, and provide an exact ULP-error definition. The final test vector demonstrates a unit/spacing trap: for 0.1f the local float32 spacing is 2^-27, so the ULP error is 0.2, not 0.1.

- The reconstruction formula is valid only for normalized finite values. It is incorrect for zero, subnormal numbers, infinities, and NaNs.

- For normalized values, exp must mean the stored exponent field E and the exponent must be E minus the format bias; this is not stated precisely.

- Subnormal reconstruction is missing: value=(-1)^s·(0.mantissa)·2^(1-bias), with no implicit leading 1.

- Zero and signed zero special cases are missing; both exponent and mantissa fields are zero, but the reconstructed value is ±0 rather than 1·2^(-bias).

- Infinity and NaN encodings, where the exponent field is all ones, are not defined.

- No formula defines ULP error, and the result depends on whether the reference is the exact decimal input, a binary value, or another float. ULP spacing also differs at subnormal values and zero.

## Units and conventions

- Bit fields, exponents, mantissas, and ULP counts are dimensionless; hexadecimal values represent raw bit patterns, not numeric magnitudes.

- The radix is 2.

- For normalized float32 values, bias=127, fraction width=23, and the spacing at unbiased exponent e is 2^(e-23).

- For normalized float64 values, bias=1023, fraction width=52, and the spacing at unbiased exponent e is 2^(e-52).

- For decimal-input vectors, the decimal is treated as an exact real number before round-to-nearest, ties-to-even conversion to the selected IEEE-754 format.

- The test-vector ULP error is defined as (reconstructed_value-exact_decimal_input)/local_spacing; this definition is not present in the supplied specification.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "decimal": 1.5,
    "precision": 64
  },
  "expected_outputs": {
    "sign_bits": "0",
    "exponent_bits": "01111111111",
    "mantissa_bits": "1000000000000000000000000000000000000000000000000000",
    "raw_hex": "0x3FF8000000000000",
    "decimal_reconstruction": 1.5,
    "ulp_error": 0
  },
  "working": "1.5 = 1 + 1/2 = 1.1₂. The unbiased exponent is 0, so E=0+1023=1023=01111111111₂. Sign s=0. The fraction field is 1000...000₂, or 0x8000000000000. Reconstruct: (+1)·(1+1/2)·2^0 = 1.5. The decimal input is exactly representable, so rounding error=1.5-1.5=0 and ULP error=0."
}
````

## Failure boundary

**BREAK_CLAIMED** · SELF_FLAGGED

The stated normal-number formula is invalid for zero, subnormals, infinities, and NaNs, and raw hexadecimal input is ambiguous without an explicit word-versus-byte-order convention.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/ieee-754-float-inspector/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/ieee-754-float-inspector/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/ieee-754-float-inspector/qualification/>)

## Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

- `VERIFIED_LEDGER.json`

- `evidence-tranche2/tranche2-results.json`

- `BOUNDARY_HAZARD_CATALOG.json`