Description
Evaluate both by Horner's method with finite intermediate checks, reject a zero denominator, and return their ratio plus exact equality with excludedPoint.
A polynomial or rational expression has coefficients, an evaluation rule, and a domain. An algebraically reduced rational form can extend across a removed hole, so evaluating that form does not by itself restore the original expression's domain.
Evaluate both by Horner's method with finite intermediate checks, reject a zero denominator, and return their ratio plus exact equality with excludedPoint.
When to use this tool
Use Rational Expression Hole Substitution when you need one bounded polynomial representation or rational-evaluation component.
How to use the tool and how it works
- Enter reducedNumeratorAscending, reducedDenominatorAscending, excludedPoint, evaluationPoint. Keep parallel arrays aligned. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read value, excludedPointEncountered. Inspect excludedPointEncountered before interpreting the associated values.
Require nonempty reduced numerator/denominator coefficient arrays in ascending degree order. Evaluate both by Horner's method with finite intermediate checks, reject a zero denominator, and return their ratio plus exact equality with excludedPoint.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| reducedNumeratorAscending | Reduced Numerator Ascending | Required. Array of finite number; maximum items 256. |
| reducedDenominatorAscending | Reduced Denominator Ascending | Required. Array of finite number; maximum items 256. |
| excludedPoint | Original-domain hole supplied by the caller; no algebraic factor cancellation is verified. | Required. Finite number. |
| evaluationPoint | Evaluation Point | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| value | Numeric value on the basis stated in the calculation above. | Required. Finite number. |
| excludedPointEncountered | Excluded Point Encountered | Required. Boolean. |
{
"input": {
"reducedNumeratorAscending": [
8,
2
],
"reducedDenominatorAscending": [
1,
1
],
"excludedPoint": 2,
"evaluationPoint": 2
},
"output": {
"value": 4,
"excludedPointEncountered": true
}
}Limitations and assumptions
- The caller must supply an already reduced expression and an actual excluded point; a returned extension value does not restore the original expression's domain or certify a removable hole.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- 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. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Track the original domain and verify any claimed cancellation or equivalence before using the reduced expression.
References
Similar or alternative tools
- Polynomial Missing Degree Padding
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- Inverse Function Local Derivative
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