5: Functions

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Questions and Answers

What is the purpose of functions in programming?

  • To encapsulate frequently used functionality (correct)
  • To manage memory allocation
  • To handle errors
  • To create complex algorithms

Procedural programming is another term for object-oriented programming.

False (B)

What does the function intervals_overlap check for?

It checks if two intervals overlap.

The program computes the n-th power of a number using a loop, where the result is stored in a variable called ______.

<p>result</p> Signup and view all the answers

Match the following programming terms with their descriptions:

<p>Function = Encapsulates a block of reusable code Procedural Programming = Structured programming based on procedures or functions Interval = A range between two numbers Loop = Repeatedly executes a block of code</p> Signup and view all the answers

What does the symbol 'β' represent in a floating-point number system?

<p>The base of the number system (A)</p> Signup and view all the answers

The normalized representation of a floating-point number can have multiple forms.

<p>False (B)</p> Signup and view all the answers

What are the four natural numbers that define a floating-point number system?

<p>β, p, emin, emax</p> Signup and view all the answers

In a floating-point representation, the number 0 has ______ normalized representation.

<p>no</p> Signup and view all the answers

Match the components of a floating-point number representation with their descriptions:

<p>d0 = Leading digit in normalized form p = Number of places in precision emin = Smallest possible exponent emax = Largest possible exponent</p> Signup and view all the answers

Which of the following best describes normalized representation in floating-point numbers?

<p>A unique form of representation with d0 not equal to 0 (B)</p> Signup and view all the answers

The notation F (β, p, emin, emax) encompasses all the floating-point numbers.

<p>True (A)</p> Signup and view all the answers

What is the smallest possible exponent in a floating-point number defined by the notation F (β, p, emin, emax)?

<p>emin</p> Signup and view all the answers

Floating-point numbers can be represented in base ______.

<p>β</p> Signup and view all the answers

How is a floating-point number structured according to its representation?

<p>± d0 d1...dp−1 × β e (C)</p> Signup and view all the answers

What does the 32-bit representation of a floating point number include?

<p>Sign bit, Exponent, Mantissa (D)</p> Signup and view all the answers

The binary system is used internally by computers to perform calculations.

<p>True (A)</p> Signup and view all the answers

What is the base used for binary representation?

<p>2</p> Signup and view all the answers

Binary representations require conversion from the _____ system.

<p>decimal</p> Signup and view all the answers

Match the components with their correct descriptions in a floating-point number:

<p>Sign bit = Indicates the sign of the number Exponent = Determines the range of values Mantissa = Represents the precision of the number Floating point = Number that can represent fractions</p> Signup and view all the answers

During conversion, what is the first step in computing the binary representation of a number?

<p>Identify the integer part (D)</p> Signup and view all the answers

The mantissa includes bits after the binary point in a floating-point number.

<p>True (A)</p> Signup and view all the answers

What representation do computers use for internal calculations?

<p>binary</p> Signup and view all the answers

The mantissa in a floating point number is represented as _____ following the exponent.

<p>1.xxxxxx</p> Signup and view all the answers

What range does the exponent cover in the standard 32-bit floating-point format?

<p>-126 to 127 (D)</p> Signup and view all the answers

What is the primary purpose of the IEEE Standard 754?

<p>To standardize floating-point number systems (D)</p> Signup and view all the answers

Single precision floating-point numbers in IEEE 754 have 32 bits in total.

<p>True (A)</p> Signup and view all the answers

What is the minimum exponent value for double precision numbers in the IEEE Standard 754?

<p>-1022</p> Signup and view all the answers

The first number in the normalized representation F ∗ (β, p, emin , emax ) is called the ______.

<p>base</p> Signup and view all the answers

How many bits are used for the significand in double precision format?

<p>52 bits (D)</p> Signup and view all the answers

In floating-point addition, the exponents of the numbers must always be the same before performing binary addition.

<p>False (B)</p> Signup and view all the answers

What is the highest exponent value for single precision floating-point numbers in IEEE 754?

<p>127</p> Signup and view all the answers

Match the following terms with their descriptions related to IEEE Standard 754:

<p>Sign bit = Indicates if the number is positive or negative Significand = Contains the significant digits of the number Exponent = Defines the scale or magnitude of the number Special values = Includes representations for 0 and infinity</p> Signup and view all the answers

Normalized representation in floating-point numbers often involves scaling to the leading digit which is ______.

<p>1</p> Signup and view all the answers

When adding binary floating-point numbers, which of the following steps is NOT needed?

<p>Convert to decimal before adding (D)</p> Signup and view all the answers

What must a function that does not have a return type of void include?

<p>A return statement (D)</p> Signup and view all the answers

The return value of a function is always converted to its return type.

<p>True (A)</p> Signup and view all the answers

What is the return type of the function 'bin_digits'?

<p>int</p> Signup and view all the answers

The function call used in programming examples is typically structured as _____(functionName)(parameters).

<p>my_fun</p> Signup and view all the answers

Flashcards

String

A sequence of characters enclosed in double quotes. Examples include: "Hello, world!" or "12345".

Function

A function is a block of code that performs a specific task. Functions improve code organization and reusability, making programs easier to understand and modify.

return Statement

The return statement is used to send a value back from a function to where it was called. It's like sending a message back to the caller, carrying the result of the function's work.

Procedural Programming

Functions are used to break down large programs into smaller, modular pieces, each performing a specific task. This makes programs easier to understand, debug, and maintain.

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Postcondition

A function's postcondition describes what is expected to be true after the function is executed. It ensures the function behaves as intended and produces the desired outcome.

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Floating-point Number System

A system for representing numbers using a base, precision, and exponent range.

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Precision (p)

The number of digits used to represent a floating-point number.

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Smallest Exponent (emin)

The smallest possible exponent value in a floating-point system.

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Largest Exponent (emax)

The largest possible exponent value in a floating-point system.

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Normalized Representation

A representation of a floating-point number where the first digit is non-zero. It ensures a unique representation.

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Floating-point Number Notation

A representation of a number in base β, using an exponent and digits.

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F(β, p, emin, emax)

The set of all possible floating-point numbers that can be represented using a specific floating-point number system.

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Underflow

The number 0 and all numbers smaller than βemin cannot be represented in normalized form. These values are underflow cases.

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Multiple Representations

Different representations for the same decimal number can occur because of the choice of exponent.

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Limitations of Floating-point Arithmetic

Floating-point arithmetic operations can produce rounding errors due to limited precision.

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Floating Point Number Representation

Represents a number using a combination of a sign, exponent, and mantissa.

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Exponent Bits

The number of bits used to represent the exponent part of a floating-point number. For example, a 32-bit floating-point number often uses an 8-bit exponent.

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Mantissa Bits

The number of bits used to represent the mantissa part of a floating-point number. It holds the digits of the number.

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Base of the number system (β)

The base of the number system used for computation within the computer. Usually, this is 2, representing the binary system.

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Input/Output Base

The number system used for input and output values, typically base 10 (decimal).

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Conversion

Converting values from one base system to another, for example, converting decimal numbers to binary for internal processing.

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Decimal to Binary Conversion

A method to convert a decimal number to its binary representation.

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Exponent Range

The range of values that can be represented by a floating-point number with a given number of bits for the exponent.

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Smallest Non-Zero Value

The smallest possible non-zero value that can be represented using a given floating-point format.

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Largest Value

The largest possible value that can be represented by a given floating-point format.

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Set of Normalized Numbers

A set of normalized numbers, represented by F*(β, p, emin, emax ), where β is the base, p is the number of digits in the significand, emin is the minimum exponent, and emax is the maximum exponent.

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Calculations with Floating-point Numbers

The process of adding two floating-point numbers involves adjusting exponents, performing binary addition on the significands, renormalizing, and rounding to the specified precision.

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The IEEE Standard 754

An international standard (IEEE 754) that defines floating-point number formats and their arithmetic operations. It is widely used in computers for representing and manipulating real numbers.

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Single Precision (float)

A 32-bit floating-point number format used to represent real numbers on computers. It allows for a smaller range of values compared to double-precision numbers.

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Double Precision (double)

A 64-bit floating-point number format used to represent real numbers on computers. It allows for a wider range of values and higher precision compared to single-precision numbers.

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Rounding in Floating-point Operations

The process of taking an exact result of an operation and rounding it to the nearest representable number within the defined precision of the floating-point system.

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Single Precision Format Breakdown

The structure of the single-precision floating-point number format in the IEEE 754 standard. It uses 32 bits, divided into one sign bit, 23 bits for the significand, and 8 bits for the exponent.

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Double Precision Format Breakdown

The structure of the double-precision floating-point number format in the IEEE 754 standard. It uses 64 bits, divided into one sign bit, 52 bits for the significand, and 11 bits for the exponent.

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Why the Specific Bit Allocations?

The reason for specific bit allocations in the single and double-precision floating-point formats (IEEE 754). They are chosen to provide a balance between precision and range of representable values, while being efficient in terms of storage and performance.

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void Function

A special type of function that does not return a value. Instead, it performs an action, like displaying output or modifying data.

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Arguments of a Function

The input values that a function takes when it is called. They provide the data the function needs to do its job.

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Return Type

Indicates the type of data that a function will return. It determines what kind of value the calling code can expect.

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Study Notes

Introduction to Computer Science

  • Course details: 252-0032, 252-0047, 252-0058
  • Authors: Manuela Fischer and Felix Friedrich
  • Department: Computer Science, ETH Zurich
  • Semester: Fall 2024

Functions

  • Functions encapsulate frequently used code.
  • They partition programs into smaller subtasks.
  • Functions are also known as procedures.
  • Using functions for programming is called procedural programming.
  • A function's declaration is T fname (T₁ pname1, T2 pname2, ..., Tn pnamen) { ... } where T is the return type, pname are formal parameters, and the body is the code block.
  • A function is declared separately from its implementation, by giving the return type, name, and parameters. The implementation gives the function body.
  • Functions that return a value are different from functions that only execute side effects (like printing to the console).
  • Pre-conditions specify what the input and environment must be before a function is executed.
  • Post-conditions specify what the output and the state of the environment must be after a function execution, given valid preconditions.

Example: Computing Powers

  • Example program computes the nth power of a number.
  • The code receives a double value for a and an integer n.
  • If n is negative, the code converts a to its reciprocal and n to its positive value to calculate a^n.
  • A for loop calculates the result.
  • The result is printed to the console.

Defining Functions

  • Formal definition: T fname (T₁ pname1, T2 pname2, ..., Tn pnamen) { ... }
  • T is the return type of the function.
  • pname are the formal parameter names.
  • body is the code block executed when the function is called.

Function Calls

  • Function calls provide functionality.
  • Arguments to a function must conform to the formal parameter types.
  • Function calls are R-values.
  • Function arguments are copied into formal parameters (pass-by-value).
  • Changes to the formal parameters do not alter the original arguments.
  • A function call is an expression (with the value returned by the function).

Evaluation of Function Calls

  • Arguments are evaluated first.
  • Formal parameters are initialized with the evaluation results.
  • The function body executes.
  • A return statement (or reaching the end of the function body) ends function execution.
  • The return value (or the implicit return) is the function call's value.

Scope of Formal Parameters

  • Formal parameters are only visible within the function.
  • Their values do not affect the original calling arguments.

Pass by Value

  • Function arguments are copied into formal parameters.
  • Changes to formal parameters do not alter the original arguments.

Pre- and Postconditions

  • Preconditions describe the state required before a function executes (e.g. e >= 0 or b != 0 for pow).
  • Postconditions describe the state after the function execution (assuming valid preconditions) (e.g. return value is be).
  • Pre- and post-conditions help with function documentation and correctness.
  • Pre- and post-conditions are used to specify the input (preconditions) and expected output (postconditions) of a function.

Assertions

  • Assertions are a tool to check pre- and post-conditions, halting execution if a condition is false.
  • Assertions are used to validate function inputs and outputs (e.g., assert (e >= 0 || b != 0).
  • Assertions can be used to verify the preconditions or post-conditions of a function.

Stepwise Refinement

  • Stepwise refinement is a program design technique.
  • Start with a high-level description of the program.
  • Divide the program into smaller, solvable, tasks.
  • Repeatedly refine the solution to smaller problems, ultimately implementing all the tasks.

Libraries

  • Libraries package similar functions together.
  • Standard libraries provide essential functions (like mathematical functions, see cmath).
  • Separate compilation is used to leverage functions from other files.
  • Functions from libraries, like pow(), are called using their namespaces (e.g., std::pow).

Name Spaces

  • Name spaces prevent naming conflicts between similar functions (e.g., pow() in std and informatik namespaces).
  • The standard namespace std is used for standard library functions.
    • Namespaces are used to group related identifiers (like functions) and prevent naming conflicts.
  • Header files (.h) contain declarations, not implementations (only function signatures)

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