262 lines
3.0 KiB
Markdown
262 lines
3.0 KiB
Markdown
# Analysis #02 — Floating Point Equality Is a Lie
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## Problem
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At interviews, you often see something like:
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> Compare two floating point numbers and determine if they are equal.
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Typical answer:
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```cpp
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if (a == b) { }
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```
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Or the “better” version:
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```cpp
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fabs(a - b) < 1e-9
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```
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Looks correct. In real engineering — not enough.
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---
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## The Real Issue
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The problem is not that `==` is bad.
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The problem is the assumption:
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> There exists one correct way to compare floating point numbers.
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There isn’t.
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---
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## Why `==` Fails
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Classic example:
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```cpp
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double a = 0.1 + 0.2;
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double b = 0.3;
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```
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`a != b`
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Reasons:
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- binary representation
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- rounding errors
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- accumulation of operations
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---
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## Why “Just Use Epsilon” Is Not Enough
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### Scale problem
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```cpp
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a = 1e9;
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b = a + 0.1;
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```
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`1e-9` is meaningless here.
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---
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### Near-zero problem
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```cpp
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a = 1e-12;
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b = 2e-12;
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```
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`1e-9` is too large.
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---
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### Meaning problem
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What are these values?
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- sensor readings?
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- computed values?
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- scaled integers?
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- geometry?
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Without context, comparison is undefined.
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---
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## What This Actually Tests
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In interviews, this checks:
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- awareness of IEEE754
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- prior exposure to the trap
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- memorized epsilon usage
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Not real engineering thinking.
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---
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## Real-World Behavior
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### Sensor example
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```cpp
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if (temperature == target)
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```
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System never stabilizes.
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Reality:
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- noise
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- quantization
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- jitter
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Solution:
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- tolerance
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- hysteresis
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- filtering
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---
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### Computation paths
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```cpp
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double x = (a + b) + c;
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double y = a + (b + c);
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```
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Different results.
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---
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### Scaled data (CAN-like)
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```cpp
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raw = 1234;
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value = raw * 0.1;
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expected = 123.4;
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```
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Comparison fails.
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---
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## Demo Code
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A full demonstration program is available:
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- examples/main.cpp
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It shows:
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- where `==` fails
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- where fixed epsilon fails
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- why scale matters
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- how noise breaks naive logic
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- why hysteresis is often required
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---
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## Example Output (fill with your own results)
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Below you can include actual output from your system.
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```text
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[insert your real output here]
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```
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Use this section to demonstrate:
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- how behavior changes with different eps values
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- how noise affects stability
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- differences between float and double
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- how hysteresis stabilizes the system
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---
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## Better Approaches
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### Absolute
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```cpp
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fabs(a - b) < abs_eps
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```
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---
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### Relative
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```cpp
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fabs(a - b) < rel_eps * max(fabs(a), fabs(b))
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```
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---
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### Combined
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```cpp
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fabs(a - b) <= max(abs_eps, rel_eps * max(fabs(a), fabs(b)))
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```
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---
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### Or Avoid Float
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Compare:
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- raw values
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- fixed-point
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- integers
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---
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### Or Change the Question
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Instead of:
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> Are values equal?
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Ask:
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> Are they close enough for the system?
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---
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## Common Mistakes
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- Using `==`
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- Fixed epsilon everywhere
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- Only relative error
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- Only absolute error
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- Blind rounding
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---
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## Key Takeaway
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Floating point comparison is not a trick.
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It is a modeling problem.
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---
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## Project Perspective
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Does this problem exist in real engineering?
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Yes.
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In interview form?
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Almost never.
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Because real systems have:
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- noise
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- scale
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- constraints
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- consequences
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And no universal epsilon.
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