Numerical Methods at work

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IEEE754 Floating-Point Explorer

This tool allows you to enter a decimal number and examine how it's represented as a floating-point value, either a 32-bit float or a 64-bit double. It breaks down each component of the floating-point format−showing you the sign bit, binary mantissa, and exponent−then calculates the corresponding stored decimal value. You can work with multiple inputs, view the exact binary form, and compare how small differences appear in the stored values. Additionally, you can print or email the analysis, making it easy to share or review your results.

IEEE754 Explorer vs. 1.1
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  • Infomation
  • Help

IEEE754 Explorer - User Guide

This tool helps you understand how decimal numbers are stored as IEEE754 floating-point values (either 32-bit floats or 64-bit doubles). You can explore how the binary representation changes as you enter different decimal numbers, and see the resulting stored value and conversion error.

Steps to Use

  1. Select a Type: Choose Float for single-precision (32-bit) or Double for double-precision (64-bit).
  2. Enter a Decimal Number: In the "Decimal Number" field, type any decimal value you want to analyze (e.g., 0.1).
  3. View the Binary Representation: Once entered, the tool displays:
    • Sign Bit: Shows whether the number is positive (+) or negative (-).
    • Binary Mantissa: The fractional part of the number in binary form.
    • Exponent: The exponent bits used by IEEE754 encoding.
  4. Actual Decimal (Stored): This shows the decimal value as stored in the chosen floating-point format. Due to rounding, it may differ slightly from your original input.
  5. Conversion Error: Displays the difference between your entered decimal and the actual stored number, along with whether it is an overestimation or underestimation.

Adding More Inputs

Click the "Add Input" button to add another set of fields. This lets you compare multiple numbers or formats side-by-side.

Printing and Emailing Results

  • Print: Click the Print button to open a printable version of the current inputs and results.
  • Email: Enter your email address when prompted and submit the form to email the results to yourself.

Other Tabs

  • Information: Shows additional details or technical info on IEEE754 representation.
  • Help: This tab shows the user guide you’re currently reading.

The Information Tab

  • Decimal: 16777218 - This is the decimal representation of the number being analyzed.
  • Classification: 16777218 - This refers to the type or category of the number within the context of the analysis. E.g. NaN, Infinity or a regular number.
  • Original Decimal: 16777218 - This value is repeated to emphasize its role as the input for the analysis and to confirm consistency throughout the process.
  • Approximation Type: Float32 - This indicates the floating-point format used to represent the number, chosen due to its balance of precision and storage efficiency for 32-bit computations.
  • Approximation Value: 16777218
  • Exact Binary Representation: true - Indicates that the number can be precisely represented in binary format without any rounding errors, assuming sufficient mantissa bit is available, ensuring accuracy in computations and data storage. If the number have a exact binary represenantion but it will not fit into the allocated space of a 32bit or 64bit float the value will be false.
  • Subnormal: false - Indicates that the number is not subnormal, meaning it is represented with a normalized mantissa in the floating-point system, which ensures full precision is retained.
  • Analysis Type: float32: mantissa=24bits, exponent=8bits - The mantissa represents the significant digits of the number, while the exponent determines the scale or magnitude by specifying the power of two applied to the mantissa.
  • Reduced Fraction: 16777218/1 - This fraction represents the number in its simplest form, providing an exact mathematical representation for easier interpretation and validation.
  • Error Fraction: 0/1 - This measures the difference between the exact value and its floating-point approximation, indicating no error in this case due to exact representation.
  • Error Magnitude: 0
  • Next Representable: The Number is not calculated for exact representation
  • IEEE754 Sign: 0. 0 indictae + and 1 indicate 1 in IEEE754.
  • IEEE754 Exponent: 10010111, 2^24 - The exponent determines the scale of the number by representing the power of two to which the mantissa is multiplied, effectively positioning the decimal point. The bit sequence is the (Raw Exponent), while the second number is the biased exponent as a power of 2.
  • IEEE754 Mantissa: 00000000000000000000001 is the mantissa bit represenation with a hidden 1 in front.
  • IEEE754 HexaDecimal: 4b800001 - This representation is important as it encodes the floating-point number in a compact, human-readable format used in debugging and data analysis, derived from the binary structure of the IEEE754 standard.

Rating and Feedback

At the bottom of the page, you can rate your experience by clicking on stars and leaving a comment. Your feedback helps improve the tool.

Corrections:
19-Dec-2024vs 1.1Improved output. That can handle situation where a decimal number is exact but can't fit into the available mantissa size of a 32bit or 64bit float
13-Dec-2024vs 1.0Initial release