Articles with the ‘Floating-point’ Tag - Exploring Binary

Numbers In App Inventor Are Stored As Floating-Point

By Rick Regan September 11th, 2014

I am exploring App Inventor, an Android application development environment for novice programmers. I am teaching it to my kids, as an introductory step towards “real” app development. While playing with it I wondered: are its numbers implemented in decimal? No, they aren’t. They are implemented in double-precision binary floating-point. I put together a few simple block programs to demonstrate this, and to expose the usual floating-point “gotchas”.

App Inventor Blocks To Display 0.1 to 17 Digits

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GCC Avoids Double Rounding Errors With Round-To-Odd

By Rick Regan January 15th, 2014

GCC was recently fixed so that its decimal to floating-point conversions are done correctly; it now calls the MPFR function mpfr_strtofr() instead of using its own algorithm. However, GCC still does its conversion in two steps: first it converts to an intermediate precision (160 or 192 bits), and then it rounds that result to a target precision (53 bits for double-precision floating-point). That is double rounding — how does it avoid double rounding errors? It uses round-to-odd rounding on the intermediate result.

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real.c Rounding Is Perfect (GCC Now Converts Correctly)

By Rick Regan December 3rd, 2013

GCC, the GNU Compiler Collection, recently fixed this eight and a half year old bug: “GCC Bugzilla – Bug 21718: real.c rounding not perfect.” This bug was the cause of incorrect decimal string to binary floating-point conversions. I first wrote about it over three years ago, and then recently in September and October. I also just wrote a detailed description of GCC’s conversion algorithm last month.

This fix, which will be available in version 4.9, scraps the old algorithm and replaces it with a call to MPFR function mpfr_strtofr(). I tested the fix on version 4.8.1, replacing its copy of gcc/real.c with the fixed one. I found no incorrect conversions.

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Gay’s strtod() Returns Zero For Inputs Just Above 2^-1075

By Rick Regan November 25th, 2013

While running some of GCC’s string to double conversion testcases I discovered a bug in David Gay’s strtod(): it converts some very small subnormal numbers incorrectly. Unlike numbers 2^-1075 or smaller, which should convert to zero under round-to-nearest/ties-to-even rounding, numbers between 2^-1075 and 2^-1074 should convert to 2^-1074, the smallest number representable in double-precision binary floating-point. strtod() correctly converts the former to 0, but it incorrectly converts the latter to 0 as well.

(Update 11/25/13: This bug has been fixed.)

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College Notebook: When I Was Taught Floating-Point

By Rick Regan November 13th, 2013

In my article “Floating-Point Questions Are Endless on stackoverflow.com” I showed examples of the many questions asked that demonstrate lack of knowledge of the most basic property of floating-point — that not all decimal values are representable in binary. In response to a reader’s comment on my article I wrote:

It would be interesting to know how it’s taught today (it’s been a very long time since I was taught it). I can’t imagine though that the person teaching it wouldn’t say — within a sentence or two of saying “floating-point” — that it “can’t represent all decimal numbers accurately”.

That prompted me to look through my box of thirty plus year old college (undergraduate) notebooks. I found notebooks for four classes in which I was taught floating-point. The notes from three of those classes confirm what I thought — that we were warned early of the decimal/binary mismatch. But in the first class of the four — the beginner’s class — it’s less clear what we were told. I’ll show you images of the relevant excerpts from my notes. (I notice I had some elements of cursive in my handwriting back then.)

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GLIBC strtod() Incorrectly Converts 2^-1075

By Rick Regan November 11th, 2013

A reader of my blog, Water Qian, reported a bug to me after reading my article “How GLIBC’s strtod() Works”. I recently tested strtod(), which was was fixed to do correct rounding in glibc 2.17; I had found no incorrect conversions.

Water tested the conversion of 2^-1075 — in retrospect an obvious corner case I should have tried — and found that it converted incorrectly to 0x0.0000000000001p-1022. That’s 2^-1074, the smallest double-precision value. It should have converted to 0, under round-to-nearest/ties-to-even rounding.

(Update 11/13/13: This bug has been fixed for version 2.19.)

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Floating-Point Questions Are Endless on stackoverflow.com

By Rick Regan November 7th, 2013

For years I’ve followed, through RSS, floating-point related questions on stackoverflow.com. Every day it seems there is a question like “why does 19.24 plus 6.95 equal 26.189999999999998?” I decided to track these questions, to see if my sense of their frequency was correct. I found that, in the last 40 days, there were 18 such questions. That’s not one per day, but still — a lot!

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How GCC Converts Decimal Literals to Floating-Point

By Rick Regan November 6th, 2013

I’ve written about two implementations of decimal string to double-precision binary floating-point conversion: David Gay’s strtod(), and glibc’s strtod(). GCC, the GNU Compiler Collection, has yet another implementation; it uses it to convert decimal floating-point literals to double-precision. It is much simpler than David Gay’s and glibc’s implementations, but there’s a hitch: limited precision causes it to produce some incorrect conversions. Nonetheless, I wanted to explain how it works, since I’ve been studying it recently. (I looked specifically at the conversion of floating-point literals in C code, although the same code is used for other languages.)

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How GLIBC’s strtod() Works

By Rick Regan October 22nd, 2013

The string to double function, strtod(), converts decimal numbers represented as strings into binary numbers represented in IEEE double-precision floating-point. Many programming environments implement their string to double conversions with David Gay’s strtod(); glibc, the GNU C Library, does not.

Like David Gay’s strtod(), glibc’s strtod() produces correctly rounded conversions. But it uses a simpler algorithm: it doesn’t have a floating-point only fast path for small inputs; it doesn’t compute a floating-point approximation to the correct result; it doesn’t check the approximation with big integers; it doesn’t adjust the approximation and recheck it; it doesn’t have an optimization for really long inputs. Instead, it handles all inputs uniformly, converting their integer and fractional parts separately, using only big integers. I will give an overview of how glibc’s strtod() works.

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GCC Conversions Are Incorrect, Architecture or Otherwise

By Rick Regan October 9th, 2013

Recently I wrote about my retesting of the gcc C compiler’s string to double conversions and how it appeared that its incorrect conversions were due to an architecture-dependent bug. My examples converted incorrectly on 32-bit systems, but worked on 64-bit systems — at least most of them. I decided to dig into gcc’s source code and trace its execution, and I found the architecture dependency I was looking for. But I found more than that: due to limited precision, gcc will do incorrect conversions on any system. I’ve constructed an example to demonstrate this.

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Correctly Rounded Conversions in GCC and GLIBC

By Rick Regan September 26th, 2013

Three years ago I wrote about how the gcc C compiler and the glibc strtod() function do some decimal to double-precision floating-point conversions incorrectly. I recently retested their conversions and found out two things: glibc’s strtod() has been fixed, and gcc’s conversion code, while still unfixed, produces correct conversions on some machines.

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A Better Way to Convert Integers in David Gay’s strtod()

By Rick Regan September 20th, 2013

A reader of my blog, John Harrison, suggested a way to improve how David Gay’s strtod() converts large integers to doubles. Instead of approximating the conversion and going through the correction loop to check and correct it — the signature processes of strtod() — he proposed doing the conversion directly from a binary big integer representation of the decimal input. strtod() does lots of processing with big integers, so the facility to do this is already there.

I implemented John’s idea in a copy of strtod(). The path for large integers is so much simpler and faster that I can’t believe it never occurred to me to do it this way. It’s also surprising that strtod() never implemented it this way to begin with.

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