How Google Designed a Smarter Calculator Than Apple

Explore a 26-minute educational video by Dr. Trefor Bazett examining how Google developed a more accurate calculator than Apple's iOS version. Learn why Apple's calculator sometimes produces errors (like 10^100+1-10^100=0) due to floating point arithmetic limitations, and discover how Google researcher Hans Boehm implemented a hybrid approach using recursive real arithmetic to overcome these issues. The video breaks down complex mathematical concepts including floating point arithmetic, dyadic rational approximations, recursive reals, and implementation challenges. Follow along as Dr. Bazett explains decidability problems and how Google's approach balances accuracy with user experience considerations. The presentation includes a detailed timeline covering comparisons between calculators, mathematical foundations, and practical implementations with a minor correction noted regarding uniqueness in dyadic rational selection.

Syllabus

0:00 Apple vs Google
0:38 Floating Point Arithmetic
4:11 Rationals and Dyadic Rational Approximations
7:23 Recursive Reals
9:48 Addition of Recursive Reals
15:05 Decidability
18:00 Implementation
24:53 Check out Brilliant.org/TreforBazett
8:42 I say we can UNIQUELY choose a dyadic rational within that error bound that has the denominator 2^n. It isn't quite unique. Take 1/3 rounding to the nearest 1/4. Both 1/4 and 2/4 are within the error range. But that's ok, you can choose either of them and ultimately all we care about is choosing something within the error bound that is as small from a memory perspective, i.e. smaller denominator as possible. I can't tell from paper, but I suspect in the implementation it spends a bit of computational time to adjudicate that 1/4 is closer than 2/4 and so chooses that one