Decimal Rounding Error Fraction
Exact decimal quantum rounding error with ties toward positive infinity; signed arithmetic uses floor rather than truncation.
Description
Exact decimal quantum rounding error with ties toward positive infinity; signed arithmetic uses floor rather than truncation.
Rounding to a decimal quantum can be performed exactly as rational arithmetic. The signed error should identify which direction the result moved.
When to use this tool
Exact decimal quantum rounding error with ties toward positive infinity; signed arithmetic uses floor rather than truncation.
- Prepare the supplied value, quantum fields for this operation.
- Compare the complete example with the representation used by your own input before carrying the result into another step.
How to use the tool and how it works
- Enter value, quantum. Use the exact text syntax, units and index conventions explained below.
- Run the tool. If input validation fails, correct the supplied fields before interpreting any result.
- Read result.rounded, result.error. Interpret false, null and empty lists using the distinctions below; they can be valid outputs.
Parse value and positive quantum exactly. Divide value by quantum, compare its remainder with half a quantum and choose the upper multiple on ties. Return reduced rational rounded and rounded-minus-value error.1
| Input field | Meaning and representation | Accepted input |
|---|---|---|
| value | value. | Text; at most 8,192 UTF-16 code units, subject to the grammar above |
| quantum | quantum. | Text; at most 8,192 UTF-16 code units, subject to the grammar above |
| Output field | Interpretation |
|---|---|
| result | Object containing the named outputs below. |
| result.rounded | The rounded value under the stated formula or text convention. |
| result.error | The error value under the stated formula or text convention. |
{
"input": {
"value": "0.25",
"quantum": "0.1"
},
"output": {
"result": {
"rounded": {
"numerator": "3",
"denominator": "10"
},
"error": {
"numerator": "1",
"denominator": "20"
}
}
}
}Limitations and assumptions
- Half ties go toward positive infinity, including negative values. No floating-point approximation or alternative tie policy is used.
- Exact decimal-spelling operations require digits before the optional decimal point and allow a signed exponent. Decimal text is bounded to 256 code units and effective scale to minus 256 through 256; integer-string fields allow a sign and up to 256 digits.
- Every declared top-level field is required and unexpected fields are rejected. Numeric inputs must be finite; ordinary text fields are bounded to 8,192 UTF-16 code units, with tighter operation-specific limits where stated.
- Lists are limited to 128 entries. JSON text must meet bounded size and depth checks, and numeric outputs beyond the safe finite magnitude are rejected. Exact large integers use decimal strings.
- Indexes are zero-based and spans have an exclusive end unless explicitly stated otherwise. A valid false, null or empty-list result is distinct from malformed input.
Alternative or Complementary methods
Preserve decimal and integer strings when exact comparisons matter. Select signedness, scale, rounding and overflow policies separately, and use the matching encoder/decoder rather than a floating-point approximation.
References
Similar or alternative tools
- Decimal Tie Rounding Classifier
Exact decimal quantum half-tie classification; signed arithmetic uses floor rather than truncation.
- Binary Fraction Grid Step
Return an exact binary fraction grid step as integer strings, including the unit step at zero fractional bits.