Tag handbookfp

Here are some concerns: Say you want to compute $a+b+c+d$ and they are all of 32 bit precision, but the machine supports 64 bit....

Posted by Beetle B. on Mon 16 September 2019

The book provides an algorithm. I didn’t bother writing the details.

Posted by Beetle B. on Fri 07 June 2019

When the condition number is not high, one can do the naive algorithm for the dot product. Otherwise, one should do the...

Posted by Beetle B. on Fri 07 June 2019

Reordering the Operands, and a Bit More General ideas: Sort all your summands in ascending order (magnitude). Even more complex, sort...

Posted by Beetle B. on Thu 23 May 2019

The problem with the previous error bounds is that they are in terms of quantities like $\sum |a_i|$, which are not known in advance,...

Posted by Beetle B. on Thu 16 May 2019

Theorem for FMA Let $x,y,z$ be nonnegative floating point numbers. Assuming underflow does not occur, then \(xy+z\le...

Posted by Beetle B. on Wed 15 May 2019

Let the rounding mode be RN. Assume no overflow occurs. Then if you do recursive summation, the following inequality related to the...

Posted by Beetle B. on Fri 10 May 2019

Unless specified otherwise, everything in this chapter assumes no underflow. We often will have many factors of $(1+{ϵ}_i)$,...

Posted by Beetle B. on Fri 10 May 2019

The error of an FMA calculation is not always a floating point number. However, we can use two floating point numbers to exactly...

Posted by Beetle B. on Fri 19 April 2019

Suppose you need to multiply by a constant that is not exactly representable. Think $\pi$ and the like. We’d like to multiply and...

Posted by Beetle B. on Fri 19 April 2019

This is a short section with some details and magic numbers. I did not bother.

Posted by Beetle B. on Fri 19 April 2019

The algorithms are in the book. I did not reproduce them here. I did not read the rest of the section. There are a lot more details there.

Posted by Beetle B. on Fri 19 April 2019

This section deals with changing bases. The most obvious application is to go back and forth between decimal and binary (to make it easy...

Posted by Beetle B. on Fri 19 April 2019

The Basic Methods One way is to use Newton’s iteration on $f(x)=x^2-a$. This method for calculating square root goes back thousands...

Posted by Beetle B. on Fri 19 April 2019

This section deals with floating point $a,b$ values, not necessarily between 1 and 2. Assume they are non-negative, though. For this,...

Posted by Beetle B. on Thu 18 April 2019

We need to calculate $o(a/b)$ where $a,b$ are binary floating point numbers, and $o$ is RN, RD, RU or RZ. We have a useful proof:...

Posted by Beetle B. on Wed 17 April 2019

Assume $\beta =2$ for this section. Some of it may not work for decimal. We want to approximate $b/a$. Assume $1\le a,b<2$. In...

Posted by Beetle B. on Tue 26 March 2019

In this section, assume $\beta =2$. Now given a floating point $x$, we want to form two floating point numbers $x_h$ and...

Posted by Beetle B. on Tue 26 March 2019

For this article, define a representable pair for a floating point number $x$ to be any pair $(M,e)$ such that...

Posted by Beetle B. on Mon 25 March 2019

The 2MultFMA Algorithm This has been covered elsewhere. It works well when you use FMA. If No FMA Is Available If there is no FMA...

Posted by Beetle B. on Thu 21 March 2019

Let $a,b$ be two floating point numbers. Let $s$ be $\text{RN}(a+b)$. Regardless of which number it picks in a tie, it can be shown that...

Posted by Beetle B. on Thu 21 March 2019

When you multiply a floating point number by a power of $\beta$, the result is exact provided there is no over or underflow. Another...

Posted by Beetle B. on Wed 20 March 2019

Sterbenz’s Lemma: If your floating point system has denormals, and if $x,y$ are non-negative, finite floating point numbers such that...

Posted by Beetle B. on Wed 20 March 2019

To get $p$ of the floating point system you are on: i = 0 A = 1.0 B = 2 # The radix. while (A + 1.0) - A == 1.0: A = B * A i += 1...

Posted by Beetle B. on Wed 20 March 2019

Suppose we want to compute the radix of a floating point system. The code below will do it for you - it works assuming the...

Posted by Beetle B. on Wed 20 March 2019

Do not assume that the operations in a programming language will map to the ones in the standard. The standard was originally written...

Posted by Beetle B. on Mon 04 March 2019

I skipped the rest of the chapter (inlcuding hardware details).

Posted by Beetle B. on Thu 07 February 2019

NaN Signaling NaNs do not appear as the result of arithmetic operations. When they appear as an operand, they signal an...

Posted by Beetle B. on Thu 07 February 2019

Invalid The default result of such an operation is a quiet NaN. The operations that lead to Invalid are: Most operations on a...

Posted by Beetle B. on Wed 06 February 2019

This section addresses how one can convert a character sequence into a decimal/binary floating point number. Decimal Character Sequence...

Posted by Beetle B. on Wed 06 February 2019

The standard requires that you can compare any two floating point numbers, as long as they share the same radix. The unordered condition...

Posted by Beetle B. on Wed 06 February 2019

Rounding Direction Attributes IEEE 754-2008 requires that the following be correctly rounded: Arithmetic operations: Addition...

Posted by Beetle B. on Wed 30 January 2019

Arithmetic Operations and Square Root Handling Signed 0 If $x,y$ are nonzero, and $x+y=0$ or $x-y=0$ exactly, then it is $+0$...

Posted by Beetle B. on Wed 30 January 2019

The standard defines several interchange formats to allow for transferring floating point data between machines. They could be as bit...

Posted by Beetle B. on Mon 28 January 2019

It has been shown that $\beta =2$ gives better worst case and average accuracy than all other bases. if...

Posted by Beetle B. on Tue 22 January 2019

Floating point addition and multiplication are still commutative. Associativity is compromised, though. An example: Let...

Posted by Beetle B. on Tue 22 January 2019

In IEEE-754, the implementer can signal an exception along with the result of the operation. Usually (or perhaps mandated?), the signal...

Posted by Beetle B. on Tue 22 January 2019

Let $o$ be the rounding function, and $a,b,c$ are floating point numbers. Then $\mathrm{FMA}(a,b,c)$ is $o(ab+c)$. if...

Posted by Beetle B. on Tue 22 January 2019

Converting From ULP Errors to Relative Errors Let $x$ be in the normal range, and $|x-X|=\alpha \text{ulp}(x)$. Then: \begin{equation*}...

Posted by Beetle B. on Tue 22 January 2019

There are multiple definitions of unit in the last place. I think most agree when $x$ is not near a boundary point. Here is the...

Posted by Beetle B. on Wed 16 January 2019

Ranges The normal range is the set of real numbers: ${\beta }^{e_{\mathit{\text{min}}}}\le |x|\le \mathrm{\Omega }$ and the subnormal range are where...

Posted by Beetle B. on Tue 15 January 2019

The IEEE 754-2008 specifies five rounding functions: Round toward $-\mathrm{\infty }$ (RD): It is the largest floating point number less than or...

Posted by Beetle B. on Mon 14 January 2019

0 (some systems have signed 0’s as well) NaN for any invalid operation $\mathrm{\infty }$ (some systems are signed, some are not). In the IEEE...

Posted by Beetle B. on Fri 11 January 2019

Underflow before rounding occurs when the absolute value of the exact value is strictly less than ${\beta }^{e_{\mathit{\text{min}}}}$ (i.e. the...

Posted by Beetle B. on Fri 11 January 2019

We would like a unique way to represent $x$. One approach is to pick the one which gives the smallest exponent possible (while still...

Posted by Beetle B. on Fri 11 January 2019

A radix $\beta$ floating point number $x$ is of the form $m\beta e$, where $|m|<\beta$ is called the significand and $e$ is...

Posted by Beetle B. on Fri 11 January 2019