Floating-point arithmetic

Abstract

Floating-point numbers have an intuitive meaning when it comes to physics-based numerical computations, and they have thus become the most common way of approximating real numbers in computers. The IEEE-754 Standard has played a large part in making floating-point arithmetic ubiquitous today, by specifying its semantics in a strict yet useful way as early as 1985. In particular, floating-point operations should be performed as if their results were first computed with an infinite precision and then rounded to the target format. A consequence is that floating-point arithmetic satisfies the ‘standard model’ that is often used for analysing the accuracy of floating-point algorithms. But that is only scraping the surface, and floating-point arithmetic offers much more.In this survey we recall the history of floating-point arithmetic as well as its specification mandated by the IEEE-754 Standard. We also recall what properties it entails and what every programmer should know when designing a floating-point algorithm. We provide various basic blocks that can be implemented with floating-point arithmetic. In particular, one can actually compute the rounding error caused by some floating-point operations, which paves the way to designing more accurate algorithms. More generally, properties of floating-point arithmetic make it possible to extend the accuracy of computations beyond working precision.

Error in Ulps of the Multiplication or Division by a Correctly-Rounded Function or Constant in Binary Floating-Point Arithmetic

“…Since c is between 1 and 2, the bound ( 8) is always less than the bound (7). Hence, the bound (7) holds in all cases. We immediately deduce…”

“…It is wiser to measure errors in terms of ulps of the exact result instead of ulps of the computed result, because the latter choice could lead to dubious conclusions. The authors of [7,Section 2.5] illustrate this as follows:…”

Optical Mode Calculation in Large-Area Photonic Crystal Surface-Emitting Lasers

“…For some structures, exponentials e σj |Sj | can be huge. To bypass computer arithmetic problems [18], we introduced bounded matrices M j = M j e −σj |Sj | and explored them to construct the characteristic function. An example of calculated vertical mode, the corresponding function n 0 (z) and n eff , is given in Fig.…”

“…( 8). Notably, analytic formulas representing G r,s and related double integrals G (k,j) (r,s) for large |r| and |s| can imply floatingnumber-arithmetic-related problems [18] since we must handle very large and small exponentials e ± √ r 2 +s 2 β0z . By treating large and small exponentials separately (i.e., replacing possibly huge matrices M p,j with bounded matrices M p,j ), avoiding division of very large and small numbers, and accounting for further computer-arithmetic problems (such as ε + 1 − 1 ≡ 0 whereas ε + (1 − 1) ≡ ε for |ε| < 10 −16 ), we could use otherwise unavailable large values of D: see, e.g., Fig.…”