Home > Articles > Programming

Hacker's Delight: The Basics

Henry S. Warren, author of "Hacker's Delight," offers timesaving techniques, algorithms, and tricks that help programmers build more elegant and efficient software. In this chapter, he starts with the basics.
This chapter is from the book

This chapter is from the book

2–1 Manipulating Rightmost Bits

Some of the formulas in this section find application in later chapters.

Use the following formula to turn off the rightmost 1-bit in a word, producing 0 if none (e.g., 01011000 ⇒ 01010000):

x & (x1)

This can be used to determine if an unsigned integer is a power of 2 or is 0: apply the formula followed by a 0-test on the result.

Use the following formula to turn on the rightmost 0-bit in a word, producing all 1’s if none (e.g., 10100111 ⇒ 10101111):

x | (x+ 1)

Use the following formula to turn off the trailing 1’s in a word, producing x if none (e.g., 10100111 ⇒ 10100000):

x & (x + 1)

This can be used to determine if an unsigned integer is of the form 2n– 1, 0, or all 1’s: apply the formula followed by a 0-test on the result.

Use the following formula to turn on the trailing 0’s in a word, producing x if none (e.g., 10101000 ⇒ 10101111):

x | (x1)

Use the following formula to create a word with a single 1-bit at the position of the rightmost 0-bit in x, producing 0 if none (e.g., 10100111 ⇒ 00001000):

¬x & (x + 1)

Use the following formula to create a word with a single 0-bit at the position of the rightmost 1-bit in x, producing all 1’s if none (e.g., 10101000 ⇒ 1111 0111):

¬x | (x1)

Use one of the following formulas to create a word with 1’s at the positions of the trailing 0’s in x, and 0’s elsewhere, producing 0 if none (e.g., 01011000 ⇒ 00000111):


The first formula has some instruction-level parallelism.

Use the following formula to create a word with 0’s at the positions of the trailing 1’s in x, and 0’s elsewhere, producing all 1’s if none (e.g., 10100111 ⇒ 11111000):

¬x | (x + 1)

Use the following formula to isolate the rightmost 1-bit, producing 0 if none (e.g., 01011000 ⇒ 00001000):

x & (−x)

Use the following formula to create a word with 1’s at the positions of the rightmost 1-bit and the trailing 0’s in x, producing all 1’s if no 1-bit, and the integer 1 if no trailing 0’s (e.g., 01011000 ⇒ 00001111):

x ⊕ (x1)

Use the following formula to create a word with 1’s at the positions of the rightmost 0-bit and the trailing 1’s in x, producing all 1’s if no 0-bit, and the integer 1 if no trailing 1’s (e.g., 01010111 ⇒ 00001111):

x ⊕ (x + 1)

Use either of the following formulas to turn off the rightmost contiguous string of 1’s (e.g., 01011100 ==> 01000000) [Wood]:


These can be used to determine if a nonnegative integer is of the form 2j − 2k for some j ≥ k≥ 0: apply the formula followed by a 0-test on the result.

De Morgan’s Laws Extended

The logical identities known as De Morgan’s laws can be thought of as distributing, or “multiplying in,” the not sign. This idea can be extended to apply to the expressions of this section, and a few more, as shown here. (The first two are De Morgan’s laws.)


As an example of the application of these formulas, ¬(x | –(x + 1)) = ¬x &¬–(x + 1) = ¬x & ((x + 1) – 1) = ¬x & x = 0.

Right-to-Left Computability Test

There is a simple test to determine whether or not a given function can be implemented with a sequence of add’s, subtract’s, and’s, or’s, and not’s [War]. We can, of course, expand the list with other instructions that can be composed from the basic list, such as shift left by a fixed amount (which is equivalent to a sequence of add’s), or multiply. However, we exclude instructions that cannot be composed from the list. The test is contained in the following theorem.

THEOREM. A function mapping words to words can be implemented with word-parallel add, subtract, and, or, and not instructions if and only if each bit of the result depends only on bits at and to the right of each input operand.

That is, imagine trying to compute the rightmost bit of the result by looking only at the rightmost bit of each input operand. Then, try to compute the next bit to the left by looking only at the rightmost two bits of each input operand, and continue in this way. If you are successful in this, then the function can be computed with a sequence of add’s, and’s, and so on. If the function cannot be computed in this right-to-left manner, then it cannot be implemented with a sequence of such instructions.

The interesting part of this is the latter statement, and it is simply the contra-positive of the observation that the functions add, subtract, and, or, and not can all be computed in the right-to-left manner, so any combination of them must have this property.

To see the “if” part of the theorem, we need a construction that is a little awkward to explain. We illustrate it with a specific example. Suppose that a function of two variables x and y has the right-to-left computability property, and suppose that bit 2 of the result r is given by

We number bits from right to left, 0 to 31. Because bit 2 of the result is a function of bits at and to the right of bit 2 of the input operands, bit 2 of the result is “right-to-left computable.”

Arrange the computer words x, x shifted left two, and y shifted left one, as shown below. Also, add a mask that isolates bit 2.


Now, form the word-parallel and of lines 2 and 3, or the result with row 1 (following Equation (1)), and and the result with the mask (row 4 above). The result is a word of all 0’s except for the desired result bit in position 2. Perform similar computations for the other bits of the result, or the 32 resulting words together, and the result is the desired function.

This construction does not yield an efficient program; rather, it merely shows that it can be done with instructions in the basic list.

Using the theorem, we immediately see that there is no sequence of such instructions that turns off the leftmost 1-bit in a word, because to see if a certain 1-bit should be turned off, we must look to the left to see if it is the leftmost one. Similarly, there can be no such sequence for performing a right shift, or a rotate shift, or a left shift by a variable amount, or for counting the number of trailing 0’s in a word (to count trailing 0’s, the rightmost bit of the result will be 1 if there are an odd number of trailing 0’s, and we must look to the left of the rightmost position to determine that).

A Novel Application

An application of the sort of bit twiddling discussed above is the problem of finding the next higher number after a given number that has the same number of 1-bits. You might very well wonder why anyone would want to compute that. It has application where bit strings are used to represent subsets. The possible members of a set are listed in a linear array, and a subset is represented by a word or sequence of words in which bit i is on if member i is in the subset. Set unions are computed by the logical or of the bit strings, intersections by and’s, and so on.

You might want to iterate through all the subsets of a given size. This is easily done if you have a function that maps a given subset to the next higher number (interpreting the subset string as an integer) with the same number of 1-bits.

A concise algorithm for this operation was devised by R. W. Gosper [HAK, item 175].1 Given a word x that represents a subset, the idea is to find the rightmost contiguous group of 1’s in x and the following 0’s, and “increment” that quantity to the next value that has the same number of 1’s. For example, the string xxx0 1111 0000, where xxx represents arbitrary bits, becomes xxx1 0000 0111. The algorithm first identifies the “smallest” 1-bit in x, with s = x &–x, giving 0000 0001 0000. This is added to x, giving r = xxx1 0000 0000. The 1-bit here is one bit of the result. For the other bits, we need to produce a right-adjusted string of n–1 1’s, where n is the size of the rightmost group of 1’s in x. This can be done by first forming the exclusive or of r and x, which gives 0001 1111 0000 in our example.

This has two too many 1’s and needs to be right-adjusted. This can be accomplished by dividing it by s, which right-adjusts it (s is a power of 2), and shifting it right two more positions to discard the two unwanted bits. The final result is the or of this and r.

In computer algebra notation, the result is y in

A complete C procedure is given in Figure 2–1. It executes in seven basic RISC instructions, one of which is division. (Do not use this procedure with x = 0; that causes division by 0.)

If division is slow but you have a fast way to compute the number of trailing zeros function ntz(x), the number of leading zeros function nlz(x), or population count (pop(x) is the number of 1-bits in x), then the last line of Equation (2) can be replaced with one of the following formulas. (The first two methods can fail on a machine that has modulo 32 shifts.)

Figure 2-1

Figure 2-1. Next higher number with same number of 1-bits.

InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.


Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.


If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information

Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.


This site is not directed to children under the age of 13.


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure

Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact

Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice

We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020