FAQ
TL;DR: On AVR, float division costs ~465 cycles and sqrt ~492; as one user put it, "That seems like an awfully long time to perform a division." Switch to fixed‑point, small‑angle series, or lookup tables to cut latency. [Elektroda, Edward Henderson, post #21657643]
Why it matters: This FAQ helps AVR flight‑control developers hit tight loop deadlines by choosing math that fits 8‑bit hardware.
Quick Facts
- AVR has 8‑bit hardware multiply, but no divide; float __divsf3 ≈ 465 cycles, sqrt ≈ 492 cycles. [Elektroda, Edward Henderson, post #21657643]
- Float sin/cos can take 1,600+ cycles on small AVRs; consider series or tables. [Elektroda, Edward Henderson, post #21657645]
- Q8.8 fixed‑point: ~40‑cycle multiply and ~360‑cycle worst‑case divide on AVRGCC. [Elektroda, Bruce Land, post #21657651]
- Simple 16‑bit reciprocal and sqrt routines exist for ATmega644 in GCC. [Elektroda, Bruce Land, post #21657652]
- Table tradeoff: 8‑bit mapping ≈ 256 bytes; 12‑bit ≈ 1,536 bytes of storage. [Elektroda, Ralph Pruitt, post #21657654]
What makes float division and sqrt slow on AVR 8‑bit?
AVR lacks a hardware divider. Library float operations must emulate divide and sqrt in software. That adds many steps. Reported costs are ~465 cycles for float divide and ~492 cycles for sqrt. Hardware multiply exists for 8‑bit integers only, so floats need extra work. Use fixed‑point or algorithmic shortcuts to avoid these costs. [Elektroda, Edward Henderson, post #21657643]
Should I switch to fixed‑point for flight‑control math?
Yes, for speed and determinism. In Q8.8, 16‑bit fixed‑point multiply runs about 40 cycles, and divide about 360 cycles worst case. That is a large gain over floating point on AVR. Fixed‑point keeps timing tight and predictable in control loops. Start by scaling sensor inputs and gains into Q formats. [Elektroda, Bruce Land, post #21657651]
How can I compute reciprocal or sqrt quickly on ATmega644?
Use lightweight 16‑bit routines. A simple reciprocal and sqrt implementation for ATmega644 in GCC is available. These avoid general‑purpose float overhead and suit control inner loops. Integrate them with Q8.8 arithmetic to keep data movement simple. Validate range and precision against your control requirements before deployment. [Elektroda, Bruce Land, post #21657652]
Are Taylor series good for sin/cos on small angles?
Yes. For small angles, a truncated Taylor series uses only adds and multiplies. That maps well to AVR hardware and runs much faster than float trig. One developer noted float sin/cos took 1,600+ cycles on their device. Keep input in radians and bound the angle to the series’ valid region. [Elektroda, Edward Henderson, post #21657645]
How many cycles do float sin/cos actually take on AVR?
Expect more than 1,600 cycles per call on small AVRs using float trig. That cost can starve fast control loops or sensor fusion updates. Replace with fixed‑point approximations, lookup tables, or CORDIC‑style methods tailored to your precision target. Profile on your exact clock and compiler settings. [Elektroda, Edward Henderson, post #21657645]
What precision do I really need for realtime control?
Define it upfront. "Find out how precise you will have to be." An 8‑bit sine or cosine can incur about 1.4% error. If that is unacceptable, move to 16‑bit or higher, or add calibration. Match numeric width to stability margins and sensor noise to avoid wasted cycles. [Elektroda, Per Zackrisson, post #21657653]
When should I use lookup tables on AVR?
Use tables when you need predictable latency and can spare memory. Precompute results and index by input to trade RAM/flash for speed. For an 8‑bit input and output, a 256‑byte table works; 12‑bit mappings can need about 1,536 bytes. Hybridize with algorithms for critical regions. [Elektroda, Ralph Pruitt, post #21657654]
How do piecewise‑linear tables cut memory without losing accuracy?
Store anchor points and linearly interpolate between them. Choose breakpoints so interpolation meets your resolution target. This reduces table size for a small compute cost. It suits monotonic functions like sqrt or atan segments in attitude math. Validate worst‑case error at the segment boundaries. [Elektroda, Todd Hayden, post #21657655]
Where should I ask about AVR math performance and tips?
AVR Freaks is a long‑running community focused on AVR hardware and GCC toolchains. Post your cycle counts, compiler flags, and target MCU. You’ll get code‑level advice, library pointers, and optimization feedback from practitioners. Share a minimal test case for best results. [Elektroda, Joe Wolin, post #21657644]
Any classic resources for clever math tricks?
Yes—HAKMEM collects compact math hacks from the early MIT AI Lab. It offers bit‑level insights useful for fast integer approximations on small MCUs. Skim it for reciprocal, root, and trig approximations to adapt to fixed‑point. Verify each trick’s range and error. [Elektroda, Joe Wolin, post #21657648]
Is there a good book on realtime math for embedded systems?
Try “Math Toolkit for Real‑Time Programming” by Jack Crenshaw. It covers practical numeric methods, approximations, and implementation guidance. Useful when porting control algorithms to constrained MCUs. Pair reading with on‑board profiling to confirm gains. Borrow or buy to build your toolkit. [Elektroda, Gary Crowell, post #21657650]
How do I migrate a loop from float to Q8.8 fixed‑point?
- Scale inputs and gains by 256, and store as int16_t.
- Replace operations with Q8.8 routines; use 32‑bit intermediates for multiply.
- Audit saturation and rounding; then profile cycles and error versus float. [Elektroda, Bruce Land, post #21657651]
Why does 8‑bit hardware multiply not speed up float math?
The hardware multiply handles 8‑bit integers. Float operations are 32‑bit and require software routines. That mismatch adds many instructions for unpacking, alignment, and normalization. Use integer or fixed‑point to leverage the multiplier efficiently. Keep data paths narrow and scaled. [Elektroda, Bill Westfield, post #21657649]
Are there open‑source AVR fixed‑point libraries?
Yes. Developers point to fixed‑point fractional libraries for AVR, such as avrfix. These offer add, multiply, divide, and common transforms optimized for 8‑bit cores. Evaluate function coverage, cycle counts, and memory before integrating into flight code. [Elektroda, Bill Westfield, post #21657649]
What edge cases can break low‑precision approaches?
Low resolution can miss control targets near limits. For example, using 8‑bit trig can introduce about 1.4% error. That may degrade stability or overshoot in tight loops. Increase word size or refine tables in sensitive ranges to mitigate risk. [Elektroda, Per Zackrisson, post #21657653]
