Project 7: Floating-Point Emulator
Emulate floating-point arithmetic and rounding behavior in software.
Quick Reference
| Attribute | Value |
|---|---|
| Difficulty | Expert |
| Time Estimate | 2 weeks |
| Language | C |
| Prerequisites | Bitwise ops, IEEE basics |
| Key Topics | mantissa, exponent, rounding |
1. Learning Objectives
- Represent floating-point numbers as sign/exponent/mantissa.
- Implement addition, multiplication, and rounding modes.
- Handle special cases (zero, inf, NaN).
- Validate against IEEE 754 behavior.
2. Theoretical Foundation
2.1 Core Concepts
- Normalization: Mantissa must be normalized after ops.
- Rounding modes: nearest-even, toward zero, etc.
- Special values: NaN, inf, subnormals.
2.2 Why This Matters
TAOCP covers arithmetic details that expose how floating-point errors arise. An emulator makes these rules explicit.
2.3 Historical Context / Background
IEEE 754 standardized floating-point behavior to ensure consistent results across hardware.
2.4 Common Misconceptions
- “Floating point is exact”: It is not.
- “Rounding happens only at the end”: It happens at every operation.
3. Project Specification
3.1 What You Will Build
A software emulator for a simplified floating-point format (e.g., 1 sign, 8 exponent, 23 mantissa) with basic arithmetic.
3.2 Functional Requirements
- Encode/decode float to bit fields.
- Implement add and multiply with rounding.
- Handle zero, inf, NaN, subnormal.
- Provide comparison against hardware float.
3.3 Non-Functional Requirements
- Correctness: Matches chosen spec.
- Transparency: Trace internal steps.
- Usability: CLI test harness.
3.4 Example Usage / Output
add(1.5, 2.25) -> 3.75
mul(1e20, 1e20) -> inf
3.5 Real World Outcome
You can see why certain operations produce NaN or inf and how rounding changes results.
4. Solution Architecture
4.1 High-Level Design
┌──────────────┐ ┌──────────────┐
│ bit fields │────▶│ arithmetic │
└──────────────┘ └──────────────┘
4.2 Key Components
| Component | Responsibility | Key Decisions |
|---|---|---|
| Encoding | sign/exponent/mantissa | fixed format |
| Arithmetic | add/mul with normalize | integer math |
| Rounding | apply chosen mode | nearest-even |
4.3 Data Structures
typedef struct {
uint32_t sign;
uint32_t exp;
uint32_t mant;
} FP32;
4.4 Algorithm Overview
Key Algorithm: Floating add
- Align exponents.
- Add/sub mantissas.
- Normalize and round.
Complexity Analysis:
- Time: O(1) per op.
- Space: O(1).
5. Implementation Guide
5.1 Development Environment Setup
gcc --version
5.2 Project Structure
project-root/
├── include/fpemu.h
├── src/fpemu.c
├── tests/
└── Makefile
5.3 The Core Question You’re Answering
“How does a floating-point unit actually compute and round values?”
5.4 Concepts You Must Understand First
- Exponent bias
- Normalization rules
- Rounding modes
5.5 Questions to Guide Your Design
- How will you handle denormals?
- Which rounding mode will you implement first?
- How will you compare results with hardware float?
5.6 Thinking Exercise
Show why adding a tiny number to a huge number can produce no change.
5.7 The Interview Questions They’ll Ask
- Why does floating-point rounding exist?
- What is NaN and how is it propagated?
- How are exponents aligned in addition?
5.8 Hints in Layers
Hint 1: Implement encode/decode first. Hint 2: Add addition with normalization. Hint 3: Add multiplication and special cases.
5.9 Books That Will Help
| Topic | Book | Chapter |
|---|---|---|
| Floating-point | TAOCP Vol 2 | Ch. 4.2 |
5.10 Implementation Phases
Phase 1: Foundation (1 week)
- Encoding/decoding and basic ops.
Phase 2: Core Functionality (1 week)
- Add rounding modes and special cases.
Phase 3: Polish & Edge Cases (3-4 days)
- Compare against IEEE 754 results.
5.11 Key Implementation Decisions
| Decision | Options | Recommendation | Rationale |
|---|---|---|---|
| Format | IEEE-754-like | single precision | familiarity |
6. Testing Strategy
6.1 Test Categories
| Category | Purpose | Examples |
|---|---|---|
| Encoding | round-trip | float -> bits -> float |
| Arithmetic | correctness | add/mul cases |
| Special cases | NaN/inf | operations with NaN |
6.2 Critical Test Cases
- Add numbers with very different exponents.
- Multiply to overflow -> inf.
- NaN propagation.
6.3 Test Data
Known IEEE-754 examples
7. Common Pitfalls & Debugging
7.1 Frequent Mistakes
| Pitfall | Symptom | Solution |
|---|---|---|
| Wrong bias | wrong values | verify exponent bias |
| Missing normalization | incorrect mantissa | normalize after ops |
| Rounding errors | mismatch with IEEE | implement guard bits |
7.2 Debugging Strategies
- Print intermediate fields.
- Compare against hardware float bit patterns.
7.3 Performance Traps
Not performance-critical; accuracy first.
8. Extensions & Challenges
8.1 Beginner Extensions
- Add subtraction and division.
8.2 Intermediate Extensions
- Implement multiple rounding modes.
8.3 Advanced Extensions
- Implement extended precision format.
9. Real-World Connections
- Numerical computing, graphics, scientific software.
10. Resources
- TAOCP Vol 2 Chapter 4.2
- IEEE 754 reference
11. Self-Assessment Checklist
- I can explain exponent alignment.
- My emulator matches IEEE results for tests.
12. Submission / Completion Criteria
Minimum: Encode/decode + add. Full: Add/mul with rounding and special cases. Excellence: Multiple formats and rounding modes.
This guide was generated from LEARN_TAOCP_DEEP_DIVE.md. For the complete learning path, see the parent directory README.