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1978, The C Programming Language (Englewood Cliffs, NJ:Prentice-Hall). [Reference for K&R “traditional” C. Later editions of this book conform tothe ANSI C standard.]Hogan, T. 1984, The C Programmer’s Handbook (Bowie, MD: Brady Communications).Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).However, this is likely to produce code where SQR(sin(x)) results in two calls tothe sine routine! You might be tempted to avoid this by storing the argument of thesquaring function in a temporary variable:28Chapter 1.Preliminaries1.3 Error, Accuracy, and StabilityComputers store numbers not with infinite precision but rather in some approximation that can be packed into a fixed number of bits (binary digits) or bytes (groupsof 8 bits).

Almost all computers allow the programmer a choice among severaldifferent such representations or data types. Data types can differ in the number ofbits utilized (the wordlength), but also in the more fundamental respect of whetherthe stored number is represented in fixed-point (int or long) or floating-point(float or double) format.A number in integer representation is exact.

Arithmetic between numbers ininteger representation is also exact, with the provisos that (i) the answer is not outsidethe range of (usually, signed) integers that can be represented, and (ii) that divisionis interpreted as producing an integer result, throwing away any integer remainder.In floating-point representation, a number is represented internally by a sign bits (interpreted as plus or minus), an exact integer exponent e, and an exact positiveinteger mantissa M . Taken together these represent the numbers × M × B e−E(1.3.1)where B is the base of the representation (usually B = 2, but sometimes B = 16),and E is the bias of the exponent, a fixed integer constant for any given machineand representation. An example is shown in Figure 1.3.1.Several floating-point bit patterns can represent the same number.

If B = 2,for example, a mantissa with leading (high-order) zero bits can be left-shifted, i.e.,multiplied by a power of 2, if the exponent is decreased by a compensating amount.Bit patterns that are “as left-shifted as they can be” are termed normalized. Mostcomputers always produce normalized results, since these don’t waste any bits ofthe mantissa and thus allow a greater accuracy of the representation.

Since thehigh-order bit of a properly normalized mantissa (when B = 2) is always one, somecomputers don’t store this bit at all, giving one extra bit of significance.Arithmetic among numbers in floating-point representation is not exact, even ifthe operands happen to be exactly represented (i.e., have exact values in the form ofequation 1.3.1). For example, two floating numbers are added by first right-shifting(dividing by two) the mantissa of the smaller (in magnitude) one, simultaneouslyincreasing its exponent, until the two operands have the same exponent.

Low-order(least significant) bits of the smaller operand are lost by this shifting. If the twooperands differ too greatly in magnitude, then the smaller operand is effectivelyreplaced by zero, since it is right-shifted to oblivion.The smallest (in magnitude) floating-point number which, when added to thefloating-point number 1.0, produces a floating-point result different from 1.0 istermed the machine accuracy m . A typical computer with B = 2 and a 32-bitwordlength has m around 3 × 10−8 .

(A more detailed discussion of machinecharacteristics, and a program to determine them, is given in §20.1.) RoughlySample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.

Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).Although we assume no prior training of the reader in formal numerical analysis,we will need to presume a common understanding of a few key concepts. We willdefine these briefly in this section..