5 float points

Instructor: Jenny Song 👩🏫¶

Table of Contents 📚¶

1. Review of C Memory Layout
2. Floating Point Representation
3. IEEE 754 Standard
4. Special Cases & Limitations
5. Examples & Practice

Review: C Memory Layout 🧠¶

+------------------+
|       Stack      | 👉 Local variables (LIFO)
+------------------+
|    Static Data   | 👉 Global variables & string literals
+------------------+
|       Code       | 👉 Machine code copy
+------------------+
|       Heap       | 👉 Dynamic storage (malloc/free)
+------------------+

• Memory Bugs often arise from stack/heap collisions (OS prevents via virtual memory).

Floating Point Representation 🌌¶

Key Ideas:¶

• Scientific Notation: Normalized form ensures one non-zero digit left of the decimal point.
• Example: 1.0 × 10⁻⁹ (normalized) vs. 0.1 × 10⁻⁸ (not normalized).
• Binary Scientific Notation:
  
  Text Only

  1.0101_{two} \times 2^4 = 10110_{two} = 22_{ten}
  

6-Bit Fixed Binary Point Example:¶

• Representation: XX.XXXX (e.g., 10.1010_{two} = 2.625_{ten}).
• Range: 0 to 3.9375 (smallest difference = 2⁻⁴ = 1/16).

IEEE 754 Floating Point Standard 🏛️¶

Single Precision (32 bits):¶

(-1)^S \times (1.\text{Significand}) \times 2^{\text{(Exponent - 127)}}

• Exponent Bias: 127 (actual exponent = Exponent field - 127).
• Implicit Leading 1: Significand assumes 1.xxx..., e.g., 1.1010....

Double Precision (64 bits):¶

• Larger significand (52 bits) and exponent bias 1023.

Special Cases & Limitations ⚠️¶

Encodings Summary:¶

Key Notes:¶

• Two Zeros: +0 and -0 (same value, different sign bits).
• Infinity: Result of overflow (e.g., division by zero).
• NaN: "Not a Number" for undefined operations.

Examples & Practice 💡¶

Example 1: Convert to Single-Precision¶

Value: -3.75
1. Sign: S = 1 (negative).
2. Binary Fraction: 3.75 = 11.11_{two} = 1.111_{two} × 2^1.
3. Exponent: 1 + 127 = 128 → 10000000_{two}.
4. Significand: 111000...0 (23 bits).

Encoding:

1 10000000 11100000000000000000000

Example 2: Decode Single-Precision¶

Hex: 0xC0A00000
1. Binary: 1 10000001 01000000000000000000000.
2. Sign: Negative (S=1).
3. Exponent: 129 - 127 = 2.
4. Significand: 1.010000... = 1.25_{ten}.
5. Value: -1.25 × 2^2 = -5.0.

Practice Problems 📝¶

1. Convert 12.375 to IEEE 754 single-precision.
   - Answer: 0x41460000.

2. Decode 0x3F800000.
   - Answer: 1.0 (S=0, Exponent=127, Significand=1.0).

Key Takeaways 🚀¶

• Floating point trades precision for range.
• IEEE 754 ensures consistency across systems.
• Special values (±0, ±∞, NaN) handle edge cases gracefully.