1. What is 64-bit Double Precision?

64-bit double precision is the most common format for floating-point numbers in modern computers. It uses 1 sign bit, 11 exponent bits, and 52 fraction bits to represent a wide range of values with about 15-17 significant decimal digits precision.

2. How Does the Calculator Work?

The calculator uses the IEEE 754 double precision formula:

Where:

• \( s \) — Sign bit (0 for positive, 1 for negative)
• \( f \) — Fraction part (52 bits, 0 to 2^52-1)
• \( e \) — Exponent part (11 bits, 0 to 2047)

Special Cases:

• When \( e = 0 \): Subnormal numbers
• When \( e = 2047 \): Infinity or NaN

3. Importance of Floating Point Representation

Details: Understanding floating-point representation is crucial for numerical computing, scientific calculations, and avoiding rounding errors in financial applications.

4. Using the Calculator

Tips: Enter the sign bit (0 or 1), fraction (0 to 4,503,599,627,370,495), and exponent (0 to 2047) to calculate the corresponding floating-point value.

5. Frequently Asked Questions (FAQ)

Q1: What is the range of representable numbers?
A: Approximately ±5.0 × 10^-324 to ±1.8 × 10^308 for normal numbers.

Q2: What is machine epsilon for double precision?
A: 2^-52 ≈ 2.22 × 10^-16, the smallest difference between 1.0 and the next representable number.

Q3: How are special values represented?
A: Infinity has e=2047 and f=0. NaN has e=2047 and f≠0.

Q4: What about rounding errors?
A: Floating-point arithmetic has inherent rounding errors due to finite precision representation.

Q5: How does this compare to 32-bit float?
A: 32-bit float has 8 exponent bits and 23 fraction bits, with less precision and smaller range.