End of Error
by John L. Gustafson · first published 2015
47% recommend47% neutral5% critical
Why HN recommends it
Explains why floating-point arithmetic gives so few guarantees in practice, with worked numerical examples, code, and history of computing methods; written for a general audience needing only high school math.
Summarised by Claude from recommending comments
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In this context, I also highly recommend John Gustafson's book, The End of Error. It is full of great examples, and at one point contrasts advances in numerical computing with improvements in other areas: "In 1970, a printer might produce something that looks like this, and take…
on “High Performance Computing: Are We Just Getting Wrong Answers Faster? (1998) [pdf]” ↗Speaking of floats, I highly recommend John Gustafson's book, The End of Error: Unum Computing: https://www.crcpress.com/The-End-of-Error-Unum-Computing/Gus... It is an amazing compendium of the many ways in which floats fail to give any guarantees in practice. In some cases,…
on “Demystifying Floating Point Precision” ↗I read the book, the End of Error by John Gustafson. He works on a lot of numerical problems right there, providing code and results. He also goes over the history of floats and numerical computation, highlighting how often the best…
on “Floating-Point Formats and Deep Learning” ↗The End of Error (http://www.amazon.com/The-End-Error-Computing-Computational/...) is the book describing them in great detail... I'm working with John to put together a publicly accessible wiki, which I hope will be…
on “Nvidia Digits DevBox” ↗The End of Error: Unum Computing - John L. Gustafson https://www.crcpress.com/The-End-of-Error-Unum-Computing/Gus...
on “Ask HN: What good books have you read lately?” ↗John Gustafson has developed several promising alternatives, such as unums, posits and valids: https://posithub.org/about I also recommend his book, The End of Error - Unum Computing: https://www.crcpress.com/The-End-of-Error-Unum-Computing/Gus... A transcript of a debate with William Kahan is here: http://www.johngustafson.net/pdfs/DebateTranscription.pdf
on “Intel Underestimates Error Bounds by 1.3 quintillion (2014)” ↗…student who knows a lot more than you, especially when they help craft the tests. Intel stuck an FPGA inside its CPU. Learn Verilog and learn to hack new CPU instructions (e.g. read Gustufson's book "The End of Error" that introduces a new kind of floating point arithmetic). Write code that outperforms BLAS code. Learn BPF (Berkeley Packet Filter) so you can do impossible things about the kernel in User mode. Make…
on “The Lesson to Unlearn” ↗This echoes the way unums work in "The End of Error". (Recommended read by the way.) Good way of thinking about ranges.
on “Indices Point Between Elements” ↗"The End of Error: Unum Computing" is written for a popular audience, not fellow mathematicians. Only high school math is needed, and it's got plenty of humor and full-color illustrations and figures, in an attempt to…
on “An Interview with the Old Man of Floating-Point (1998)” ↗i recommend the book "The End of Error: Unum Computing"
on “More floating point alternatives” ↗I'd never heard of him. He's done a lot of interesting stuff, mostly in high performance computing. From his personal page: "Gustafson has recently finished writing a book, The End of Error: Unum Computing, that presents a new approach to computer arithmetic: the unum. The universal number, or unum format, encompasses all IEEE floating-point formats as well as fixed-point and exact integer…
on “Ways to Fool the Masses Giving Performance Results on Vector Computers (1991)” ↗…I could do with it until I can show myself that it works for basic arithmetic, etc, as well as the author claims. Gustafson also makes the code available (I think [here](https://www.crcpress.com/The-End-of-Error-Unum-Computing/Gus...). It's Mathematica code... there is a free viewer for that format (if you don't have Mathematica) which can print out a PDF with richly formatted equations. Also, Googling for that…
on “Unum Computing: An Energy Efficient and Massively Parallel Approach to Numerics” ↗I worked on interval arithmetic and interval arithmetic-like alternatives to floating point (I was the first independent implementor of gustafson's "the end of error"). They get hairy really fast, and that's why ultimately when it came time to re-rethink floating point, we settled again on something with fixed bit with and inexact results.
on “Yet another alternative to floating-point numbers” ↗…players it seems, which is a source of frustration for me. Why, Intel, Why?). Imaging being able to build your own instructions "on the fly". I want to implement Gustafson's UNUM arithmetic (see "The End of Error" https://www.amazon.com/End-Error-Computing-Chapman-Computati...) It's not clear what the long-term shakeout will be but it is fun to try to learn what the leading edge is doing. Like the Red Queen…
on “A Dozen Microcomputers” ↗Not yet released, but I got to read a preprint of The End of Error: Unum Computing by John L. Gustafson. There are some slides of his online which introduce the concept of Unums.
on “Ask HN: What book changed your life in 2014?” ↗