Description
Return cosine=dot/(firstNorm*secondNorm), scaledCosine=sign(firstScale*secondScale)*cosine, and their residual.
A similarity term depends on the comparison universe, missing-value policy, treatment of repeated features, and normalization. A score can also have a caller-selected value when its mathematical denominator vanishes. Those conventions matter before interpreting a threshold or comparing methods.
Return cosine=dot/(firstNorm*secondNorm), scaledCosine=sign(firstScale*secondScale)*cosine, and their residual.
When to use this tool
Use Cosine Norm Rescale Invariance Residual when you need this particular comparison term, normalization, domain condition, or empty-result policy in a larger matching workflow.
How to use the tool and how it works
- Enter dot, firstNorm, secondNorm, firstScale, secondScale. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read cosine, scaledCosine, residual. Use the output convention below.
Return cosine=dot/(firstNorm*secondNorm), scaledCosine=sign(firstScale*secondScale)*cosine, and their residual. Require positive norms, nonzero scales, and feasible supplied dot magnitude.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| dot | Dot | Required. Finite number. |
| firstNorm | First Norm | Required. Finite number; strictly above 0. |
| secondNorm | Second Norm | Required. Finite number; strictly above 0. |
| firstScale | First Scale | Required. Finite number. |
| secondScale | Second Scale | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| cosine | Cosine | Required. Finite number. |
| scaledCosine | Cosine times sign of the computed scale product; extreme underflow can affect it. | Required. Finite number. |
| residual | Signed discrepancy under the formula stated above. | Required. Finite number. |
{
"input": {
"dot": 3,
"firstNorm": 3,
"secondNorm": 2,
"firstScale": 2,
"secondScale": 4
},
"output": {
"cosine": 0.5,
"scaledCosine": 0.5,
"residual": 0
}
}Limitations and assumptions
- The scales' product is evaluated numerically and can underflow to zero despite nonzero factors. No vectors are rescaled and no robust extreme-scale invariance guarantee is provided.
- The field tables state structural bounds; the mechanics also specify cross-field constraints. Malformed inputs and unsupported domains cause errors. A successful false flag, empty branch value, or zero placeholder has the meaning stated above and is distinct from an input error.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Signed zero is exported as zero.
Alternative or Complementary methods
Compare methods only after aligning feature definitions, missingness, empty policies, multiplicities, and normalizers. Use a complete comparison when the tool returns only one contribution or count check.
References
Similar or alternative tools
- Cosine Orthogonal Residual
Inspect a supplied normalized dot product’s deviation from orthogonality; no vectors are inferred.
- Cosine Rescale Roundoff Envelope
Audit a measured cosine-rescaling discrepancy against a caller-supplied absolute roundoff budget.