Fixed Point Multiply and Rescale
Exact raw fixed-point multiplication and rescaling, truncating toward zero while retaining signed remainder evidence.
Description
Exact raw fixed-point multiplication and rescaling, truncating toward zero while retaining signed remainder evidence.
Multiplying two stored fixed-point integers introduces an extra scale factor that must be removed.
When to use this tool
Exact raw fixed-point multiplication and rescaling, truncating toward zero while retaining signed remainder evidence.
- Prepare the supplied left, right, fractionalBits 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 left, right, fractionalBits. 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.integer, result.remainder, result.divisor. Interpret false, null and empty lists using the distinctions below; they can be valid outputs.
Parse exact left/right integers and divide their product by 2^fractionalBits. Return the truncation-toward-zero quotient, signed remainder and scale divisor as decimal strings.1
| Input field | Meaning and representation | Accepted input |
|---|---|---|
| left | Exact decimal integer text; keep it quoted in JSON rather than converting through a floating-point number. | Optional leading + or - and 1 to 256 decimal digits; no whitespace |
| right | Exact decimal integer text; keep it quoted in JSON rather than converting through a floating-point number. | Optional leading + or - and 1 to 256 decimal digits; no whitespace |
| fractionalBits | fractional bits. | Integer from 0 to 128 inclusive |
| Output field | Interpretation |
|---|---|
| result | Object containing the named outputs below. |
| result.integer | Exact decimal integer text under the operation described above. |
| result.remainder | The remainder value under the stated formula or text convention. |
| result.divisor | The divisor value under the stated formula or text convention. |
{
"input": {
"left": "6",
"right": "3",
"fractionalBits": 2
},
"output": {
"result": {
"integer": "4",
"remainder": "2",
"divisor": "4"
}
}
}Limitations and assumptions
- The residual is not silently rounded. Both inputs are assumed to share the declared scale; no word-width overflow or saturation is modeled.
- 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
- Fixed Point Division and Rescale
Exact raw fixed-point division and rescaling, truncating toward zero while retaining signed remainder evidence.
- Fixed Point Add Overflow Flag
Add exact raw fixed-point integers and flag explicit signed or unsigned word overflow; does not wrap or saturate.