# Quadruple Precision, 128-bit Floating Point Arithmetic9

Posted by Cleve Moler,

The floating point arithmetic format that occupies 128 bits of storage is known as binary128 or quadruple precision. This blog post describes an implementation of quadruple precision programmed entirely in the MATLAB language.... read more >>

# “Half Precision” 16-bit Floating Point Arithmetic5

Posted by Cleve Moler,

The floating point arithmetic format that requires only 16 bits of storage is becoming increasingly popular. Also known as half precision or binary16, the format is useful when memory is a scarce resource.... read more >>

# Bank Format and Metric Socket Wrenches2

Posted by Cleve Moler,

A report about a possible bug in format bank and a visit to a local hardware store made me realize that doing decimal arithmetic with binary floating point numbers is like tightening a European bolt with an American socket wrench.... read more >>

# Ulps Plots Reveal Math Function Accuracy2

Posted by Cleve Moler,

"ULP" stands for "unit in the last place." An ulps plot samples a fundamental math function such as $\sin{x}$, or a more esoteric function like a Bessel function. The samples are compared with more accurate values obtained from a higher precision computation. A plot of the accuracy, measured in ulps, reveals valuable information about the underlying algorithms.... read more >>

# The Pentium Papers — My First MATLAB Central Contribution

Posted by Cleve Moler,

MATLAB Central is celebrating its 15th birthday this fall. In honor of the occasion, MathWorks bloggers are reminiscing about their first involvement with the Web site. My first contribution to the File Exchange was not MATLAB software, but rather a collection of documents that I called the Pentium Papers. I saved this material in November and December of 1994 when I was deeply involved in the Intel Pentium Floating Point Division Affair…. read more >>

# Iterative Refinement for Solutions to Linear Systems4

Posted by Cleve Moler,

Iterative refinement is a technique introduced by Wilkinson for reducing the roundoff error produced during the solution of simultaneous linear equations. Higher precision arithmetic is required for the calculation of the residuals.... read more >>

# Floating Point Denormals, Insignificant But Controversial

Posted by Cleve Moler,

Denormal floating point numbers and gradual underflow are an underappreciated feature of the IEEE floating point standard. Double precision denormals are so tiny that they are rarely numerically significant, but single precision denormals can be in the range where they affect some otherwise unremarkable computations. Historically, gradual underflow proved to be very controversial during the committee deliberations that developed the standard. ... read more >>

# Floating Point Numbers5

Posted by Cleve Moler,

This is the first part of a two-part series about the single- and double precision floating point numbers that MATLAB uses for almost all of its arithmetic operations. (This post is adapted from section 1.7 of my book Numerical Computing with MATLAB, published by MathWorks and SIAM.) ... read more >>