Floating-Point Calculator
IEEE 754 floating-point number converter and analyzer
IEEE 754 Floating-Point Converter
Enter any real number (supports scientific notation like 1.23e-4)
Results
Example Conversions
27.75 (Single Precision)
Binary: 01000001101111100000000000000000
Sign: 0 (positive)
Exponent: 10000011 (131 - 127 = 4)
Fraction: 01111100000000000000000
Formula: (-1)⁰ × 2⁴ × 1.734375 = 27.75
0.1 Precision Loss
Input: 0.1
Stored as: 0.10000000149011612
Error: 1.49 × 10⁻⁹
Reason: 0.1 cannot be exactly represented in binary
Special Values
+∞: 01111111100000000000000000000000
-∞: 11111111100000000000000000000000
NaN: 01111111100000000000000000000001
+0: 00000000000000000000000000000000
IEEE 754 Format
Single (32-bit)
• 1 bit: Sign
• 8 bits: Exponent
• 23 bits: Fraction
• Bias: 127
• Range: ±3.4 × 10³⁸
Double (64-bit)
• 1 bit: Sign
• 11 bits: Exponent
• 52 bits: Fraction
• Bias: 1023
• Range: ±1.7 × 10³⁰⁸
Special Cases
Zero
E=0, F=0
Positive or negative zero
Infinity
E=max, F=0
Result of overflow
NaN
E=max, F≠0
Not a Number
Subnormal
E=0, F≠0
Very small numbers
Key Concepts
Sign bit: 0 = positive, 1 = negative
Exponent is biased (subtract bias to get true exponent)
Fraction has implicit leading 1 (except subnormals)
Not all decimal numbers can be exactly represented
Double precision provides more accuracy than single
Understanding IEEE 754 Floating-Point
What is Floating-Point?
Floating-point is a standardized method for representing real numbers in computers. The IEEE 754 standard defines how these numbers are stored in binary format, allowing computers to perform mathematical operations on fractional numbers.
Why Use Floating-Point?
- •Represents a wide range of numbers (very small to very large)
- •Efficient storage in fixed number of bits
- •Hardware-optimized arithmetic operations
- •Standardized across different systems
Format Structure
S | EEEEEEEE | FFFFFFFFFFFFFFFFFFFFFFF
Sign | Exponent | Fraction (Single Precision)
Conversion Formula
(-1)^S × 2^(E-Bias) × (1.F)
S: Sign bit (0 or 1)
E: Biased exponent
F: Fractional part
Bias: 127 (single) or 1023 (double)
Precision Limitations
Floating-point numbers cannot represent all real numbers exactly. This is because:
Limited Precision
Only a finite number of bits are available for the fraction, so some decimal numbers like 0.1 cannot be represented exactly in binary.
Rounding Errors
When converting from decimal to binary and back, small rounding errors accumulate, especially in repeated calculations.