text:
Who Dat Girl
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Jump to: navigation, search
"Who Dat Girl"
Single by Flo Rida featuring Akon
from the album Only One Flo (Part 1)
Released
January 11, 2011
Format
Digital download
Recorded
2010;
Conway Recording Studios and Dr. Luke's
(Los Angeles, California)
Genre
Electro-R&B[1]
hip hop[2]
Length
3:19
Label
Atlantic
Poe Boy
Songwriter(s)
Flo Rida
Dr. Luke
Claude Kelly
Benny Blanco
Bruno Mars
Philip Lawrence
Producer(s)
Dr. Luke
Benny Blanco
Flo Rida singles chronology
"Turn Around (5, 4, 3, 2, 1)"
(2010) "Who Dat Girl"
(2011) "Dance with Me"
(2011)
"Turn Around (5, 4, 3, 2, 1)"
(2010) "Who Dat Girl"
(2011) "Dance with Me"
(2011)
Akon singles chronology
"I Just Had Sex"
(2010) I Just Had Sex2010 "Who Dat Girl"
(2011) Who Dat Girl2011 "Boomerang"
(2011) Boomerang2011
"Who Dat Girl" is a song by American rapper Flo Rida. It features American singer Akon. The song was written by Flo Rida, Dr. Luke, Claude Kelly, Benny Blanco, Bruno Mars, and Philip Lawrence, and it was produced by Dr. Luke and Benny Blanco. It was released on January 11, 2011 as the third single from Flo Rida's third studio album, Only One Flo (Part 1). It peaked at number twenty-nine on the Billboard Hot 100.
Contents
[hide]
1 Chart performance
2 Music video
3 Critical response
4 Personnel
5 Charts and certifications
5.1 Weekly charts
5.2 Year-end charts
5.3 Certifications
6 References
7 External links
Chart performance[edit]
The song debuted at number fifty-five on the Billboard Hot 100 on the issue dated November 18, 2010. On the issue date of January 22, 2011, the song charted at number thirty-nine on the chart, selling 63,000 digital copies.[3] The following week the song peaked at number twenty-nine, becoming his eighth top-thirty single.[4] In Australia, the track debuted at No. 29 on April 4, 2011,[5] and peaked at No. 10 in its fifth week on the chart.[6]
Music video[edit]
The music video was directed by Ray Kay.[7] It premiered on December 2, 2010 on MTV, with an cameo appearance by Sandra Bullock.[8] He shows his friend a picture of a girl on his laptop, and the friend accidentally spills liquid on it. The girl pops out of the laptop and throws a party at their house, like in the film Weird Science. In the video, Flo-Rida and Akon seem to be at a club scene, in which they are dancing with a few fairly good-looking ladies, covered in some substance in which looks like glow in the dark paint.[9]
Critical response[edit]
Lewis Corner of Digital Spy UK gave the song a negative review, giving the single 2 out of 5 stars, stating "Flo Rida's chart luck has been down of late, what with his last effort – and first solo track – just missing out on a spot in the top 40. His solution? Call in the big guns – in this instance Akon to launch him back to the top – and given that Flo considers himself the old-fashioned romantic sort at heart, there are no prizes for guessing which rapper took the reigns [sic] on his latest effort.
What's more, he's even roped in serial hitmaker Dr Luke to helm the track, who supplies a ready-made rubbery electro-R&B hook with a beat peeled off the 'California Gurls' cutting room floor. 'I imagine her topless/ she might set off my rocket,' Rida announces with a conviction that would make even Hugh Hefner blush, as Akon confesses his lass 'ain't no actress/ But she makes movies.' Who said romance was dead?"[10]
Personnel[edit]
Songwriting – Flo Rida, Dr. Luke, Claude Kelly, Benjamin Levin, Bruno Mars, Philip Lawrence
Production, instruments and programming – Dr. Luke, Benny Blanco
Vocals – Flo Rida, Akon
Engineering – Emily Wright, Chris "Tek" O'Ryan, Sam Holland
Assistant engineering – Tatiana Gottwald
Mixing – Serban Ghenea
Mix engineer – John Hanes
Assistant mix engineer – Tim Roberts
Background vocals – Claude Kelly
Remix Package – 8barz 8barz Music, LLC
Source:[11]
Charts and certifications[edit]
Weekly charts[edit]
Chart (2011)
Peak
position
Australia (ARIA)[12] 10
Austria (Ö3 Austria Top 40)[13] 45
Belgium (Ultratip Flanders)[14] 17
Canada (Canadian Hot 100)[15] 16
Germany Airplay (Nielsen Company)[16] 35
UK R&B (Official Charts Company)[17] 22
UK Singles (Official Charts Company)[18] 64
US Billboard Hot 100[19] 29
US Hot R&B/Hip-Hop Songs (Billboard)[20] 65
US Mainstream Top 40 (Billboard)[21] 23
US Rhythmic (Billboard)[22] 21
Year-end charts[edit]
Chart (2011)
Position
Australia (ARIA)[23] 83
Certifications[edit]
Region
Certification
Certified units/Sales
Australia (ARIA)[24]
Platinum 70,000^
Canada (Music Canada)[25]
Platinum 80,000^
*sales figures based on certification alone
^shipments figures based on certification alone
References[edit]
Jump up ^ Flo Rida feat. Akon: 'Who Dat Girl' – Music Singles Review – Digital Spy
Jump up ^ Who Dat Girl's actual genre
Jump up ^ Keith Caulfield and Silvio Pietroluongo (January 13, 2011). "Chart Moves: Edward Sharpe, Usher, Gwyneth Paltrow, Britney Spears". Billboard. Prometheus Global Media. Retrieved February 5, 2011.
Jump up ^ Cite error: The named reference sc_Billboardhot100_Flo_Rida was invoked but never defined (see the help page).
Jump up ^ Steffen Hung. "Australian charts portal". australian-charts.com. Retrieved 2012-03-15.
Jump up ^ Steffen Hung. "Flo Rida feat. Akon – Who Dat Girl". australian-charts.com. Retrieved 2012-03-15.
Jump up ^ "Flo Rida's 'Who Dat Girl' Music Video". MTV. December 2, 2010. Retrieved January 26, 2011.
Jump up ^ Rodriguez, Jayson (December 2, 2010). "Flo Rida's 'Who Dat Girl' Video Premieres". MTV. Retrieved January 26, 2011.
Jump up ^ "Flo Rida – Who Dat Girl ft. Akon [Official Video". YouTube. December 2, 2010. Retrieved 2012-03-15.
Jump up ^ "Flo Rida feat. Akon: 'Who Dat Girl' – Music Singles Review". Digital Spy. March 23, 2011. Retrieved 2012-03-15.
Jump up ^ Only One Flo (Part 1) (liner notes). Flo Rida. Atlantic Records. 2010.
Jump up ^ "Australian-charts.com – Flo Rida feat. Akon – Who Dat Girl". ARIA Top 50 Singles. Retrieved April 24, 2012.
Jump up ^ "Austriancharts.at – Flo Rida feat. Akon – Who Dat Girl" (in German). Ö3 Austria Top 40. Retrieved April 15, 2011.
Jump up ^ "Ultratop.be – Flo Rida feat. Akon – Who Dat Girl" (in Dutch). Ultratip. Retrieved April 15, 2011.
Jump up ^ "Flo Rida Chart History (Canadian Hot 100)". Billboard. Retrieved April 24, 2012.
Jump up ^ German Airplay Chart, Retrieved on 2011-06-03
Jump up ^ "Official R&B Singles Chart Top 40". Official Charts Company. Retrieved April 24, 2012.
Jump up ^ "Official Singles Chart Top 100". Official Charts Company. Retrieved April 24, 2012.
Jump up ^ "Flo Rida Chart History (Hot 100)". Billboard. Retrieved April 24, 2012.
Jump up ^ "Flo Rida Chart History (Hot R&B/Hip-Hop Songs)". Billboard. Retrieved April 24, 2012.
Jump up ^ "Flo Rida Chart History (Pop Songs)". Billboard. Retrieved April 24, 2012.
Jump up ^ "Flo Rida Chart History (Rhythmic)". Billboard. Retrieved April 24, 2012.
Jump up ^ "ARIA Top 100 Singles 2011". Australian Recording Industry Association. Retrieved January 18, 2015.
Jump up ^ "ARIA Charts – Accreditations – 2011 Singles". Australian Recording Industry Association.
Jump up ^ "Canadian single certifications – Flo Rida – Who Dat Girl". Music Canada.
External links[edit]
Lyrics of this song at MetroLyrics
[show]
v
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Flo Rida
Discography
Studio albums
Mail on Sunday
R.O.O.T.S.
Only One Flo (Part 1)
Wild Ones
EPs
Good Feeling
My House
Singles
"Low"
"Elevator"
"In the Ayer"
"Right Round"
"Shone"
"Sugar"
"Jump"
"Be on You"
"Club Can't Handle Me"
"Turn Around (5, 4, 3, 2, 1)"
"Who Dat Girl"
"Good Feeling"
"Wild Ones"
"Whistle"
"I Cry"
"Sweet Spot"
"Let It Roll"
"G.D.F.R."
"I Don't Like It, I Love It"
"My House"
Other songs
"Can't Believe It"
"How I Feel"
"Hello Friday"
"Zillionaire"
Featured singles
"Move Shake Drop"
"Starstruck"
"We Break the Dawn"
"Running Back"
"Feel It"
"Ruff Me Up"
"Bad Boys"
"Kick It Out"
"Heartbreaker"
"Good Girls Go Bad (Frank E Remix)"
"Feel It"
"iYiYi"
"You Make the Rain Fall"
"Boom Shacka"
"Higher"
"Dance with Me"
"Where Them Girls At"
"I'm Alright"
"Club Rocker"
"Hangover"
"Get Low"
"Goin' In"
"Troublemaker"
"Change Your Life"
"Say You're Just a Friend"
"Tonight Belongs to U!"
"Greenlight"
Tours
I Am... Tour
Planet Pit World Tour
[show]
v
t
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Akon
Discography
Awards and nominations
Trouble
"Locked Up"
"Ghetto"
"Lonely"
"Belly Dancer (Bananza)"
"Pot of Gold"
Konvicted
"Smack That"
"I Wanna Love You"
"Don't Matter"
"Mama Africa"
"Sorry, Blame It on Me"
"Never Took the Time"
"I Can't Wait"
Freedom
"Right Now (Na Na Na)"
"I'm So Paid"
"Beautiful"
"We Don't Care"
Extended plays
Operations of Nature
Featured singles
"Find Us"
"Moonshine"
"Slow Wind (Remix)"
"Soul Survivor"
"Snitch"
"I Am Not My Hair"
"Girls"
"The Sweet Escape"
"I Tried"
"We Takin' Over"
"Bartender"
"The Way She Moves"
"Speaker"
"Sweetest Girl (Dollar Bill)"
"Am I Dreaming"
"Hypnotized"
"Get Buck in Here"
"I'll Still Kill"
"Wanna Be Startin' Somethin' 2008"
"What You Got"
"Dangerous"
"Frozen"
"Get Low Wit It"
"Out Here Grindin"
"Wake It Up"
"Body on Me"
"What's Love"
"Day Dreaming"
"Stuck with Each Other"
"Silver & Gold"
"One"
"Just Go"
"Overtime"
"All Up 2 You"
"Sexy Bitch"
"Let's Get Crazy"
"On Top"
"Shut It Down"
"Push Push"
"Body Bounce"
"Move That Body"
"Hold My Hand"
"Kush"
"I Just Had Sex"
"Who Dat Girl"
"Boomerang"
"Chop My Money"
"Like Money"
"Play Hard"
"Holiday"
"Heatwave"
Promotional singles
"Holla Holla"
"Troublemaker"
"Dirty Work"
Other songs
"Crank It Up"
"Oh Africa"
Book
Category
Retrieved from "https://en.wikipedia.org/w/index.php?title=Who_Dat_Girl&oldid=781974785"
Categories:
2010 singles
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text:
This Little Piggy
From Wikipedia, the free encyclopedia
Jump to: navigation, search
For the Family Guy episode, see This Little Piggy (Family Guy).
"This Little Piggy"
Illustration by Lilly Martin Spencer, 1857
Nursery rhyme
Published
1760
Songwriter(s)
Unknown
"This Little Piggy" or "This Little Pig" is an English language nursery rhyme and fingerplay. It has a Roud Folk Song Index number of 19297.
Contents
[hide]
1 Lyrics
2 Finger play
3 Origins
4 In today's culture
5 See also
6 References
Lyrics[edit]
Children playing This Little Pig.[1]
The most common modern version is:
Words
Fingerplay
This little piggy went to market,
This little piggy stayed home,
This little piggy had roast beef,
This little piggy had none,
And this little piggy cried "wee wee wee" all the way home.[2]
Wiggle the "big" toe
Wiggle the "long" toe
Wiggle the "middle" toe
Wiggle the "ring" toe
Wiggle the "pinky" toe and tickle the bottom of the foot
Finger play[edit]
"... This little piggy had roast beef..."
The rhyme is usually counted out on an infant or toddler's toes, each line corresponding to a different toe, usually starting with the big toe and ending with the little toe. A foot tickle is usually added during the "Wee...all the way home" section of the last line.[citation needed] The rhyme can also be seen as a counting rhyme, although the number of each toe (from one for the big toe to five for the little toe) is never stated.[citation needed]
Origins[edit]
In 1728, the first line of the rhyme appeared in a medley called "The Nurse's Song". The first known full version was recorded in The Famous Tommy Thumb's Little Story-Book, published in London about 1760.[3]
The full rhyme continued to appear, with slight variations, in many late 18th and early 19th century collections. Until the mid-20th century, the lines referred to "little pigs".[3]
In today's culture[edit]
In 2013, an Australian children's music group, The Wiggles, wrote a song "This Little Piggy Went to Market" with the same lyrics as the original This Little Piggy with different background music. In the same year, BilinguaSing released a bilingual version of the song in English and Spanish. The song is on The Wiggles' 2014 release Apples and Bananas (DVD and CD), and the song is sung with the American journalist Lee Hawkins.[citation needed]
On Tuesday, April 29, 2014, on The Today Show, Matt Lauer played This Little Piggy on Savannah Guthrie's toes.[4]
On Monday, February 23, 2015, singer Christina Aguilera sang the rhyme in the voice of Britney Spears on "The Tonight Show starring Jimmy Fallon" in the tune and instrumental of ...Baby One More Time.[5]
See also[edit]
Children's literature portal
References[edit]
Wikisource has original text related to this article:
This Little Pig
Jump up ^ Wentworth, George; Smith, David Eugene (1912). Work and Play with Numbers. Boston: Ginn & Company. p. 14.
Jump up ^ Herman, D. (2007). The Cambridge Companion to Narrative. Cambridge: Cambridge University Press. p. 9.
^ Jump up to: a b Opie, I.; Opie, P. (1951). The Oxford Dictionary of Nursery Rhymes (1997 ed.). Oxford University Press. pp. 349–50.
Jump up ^ Duerson, Meena (April 29, 2014). "Savannah, Matt play 'This Little Piggy' to teach anchor a lesson". Today. Retrieved February 12, 2016.
Jump up ^ Boardman, Madeline (24 February 2015). "Christina Aguilera Does Incredible Britney Spears Impression on The Tonight Show With Jimmy Fallon". US Weekly. Archived from the original on 25 February 2015. Retrieved 1 June 2015.
[hide]
v
t
e
Nursery rhymes (list)
"A-Tisket, A-Tasket"
"Baa, Baa, Black Sheep"
"Did You Ever See a Lassie?"
"Hey Diddle Diddle"
"Hickory Dickory Dock"
"Hot Cross Buns"
"Humpty Dumpty"
"I'm a Little Teapot"
"It's Raining, It's Pouring"
"Jack and Jill"
"Little Bo Peep"
"Little Boy Blue"
"Little Miss Muffet"
"Mary Had a Little Lamb"
"Mary, Mary, Quite Contrary"
"Old King Cole"
"Old MacDonald Had a Farm"
"Old Mother Hubbard"
"Old Woman Who Lived in a Shoe"
"Peter Piper"
"Polly Put the Kettle On"
"Poor Mary"
"Pop Goes the Weasel"
"Pussy Cat Pussy Cat"
"Rain Rain Go Away"
"Roses Are Red"
"Rub-a-dub-dub"
"Simple Simon"
"Sticks and Stones"
"The Grand Old Duke of York"
"The Queen of Hearts"
"Twinkle, Twinkle, Little Star"
"Two Little Dickie Birds"
Clapping
"A Sailor Went to Sea"
"Down Down Baby"
"Mary Mack"
"Miss Susie"
"Miss Lucy"
"Pat-a-cake"
"Pretty Little Dutch Girl"
"Pease Porridge Hot"
Counting
"Duck, duck, goose"
"Eeny, meeny, miny, moe"
"One, Two, Buckle My Shoe"
"One, Two, Three, Four, Five"
"This Old Man"
"Tinker, Tailor"
Finger-
plays
"Five Little Monkeys"
"Itsy Bitsy Spider"
"Little Robin Redbreast"
"Round and round the garden"
"This Little Piggy"
Games
"London Bridge"
"The Muffin Man"
"Mulberry Bush"
"Oranges and Lemons"
"Ring a Ring o' Roses"
"The Farmer in the Dell"
Riddle
"As I was going to St Ives"
Lullabies
"Hush, Little Baby"
"Rock-a-bye Baby"
Rounds
"Ding Dong Bell"
"Frère Jacques"
"Kookaburra"
"Row, Row, Row Your Boat"
"Scotland's Burning"
"Three Blind Mice"
Retrieved from "https://en.wikipedia.org/w/index.php?title=This_Little_Piggy&oldid=797838338"
Categories:
Fictional pigs
Songs about pigs
Finger plays
English nursery rhymes
Roud Folk Song Index songs
Year of song unknown
Songwriter unknown
Hidden categories:
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Articles with unsourced statements from August 2013
Articles with unsourced statements from January 2016
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This page was last edited on 29 August 2017, at 13:21.
Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
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IEEE 754-1985
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IEEE 754-1985 was an industry standard for representing floating-point numbers in computers, officially adopted in 1985 and superseded in 2008 by IEEE 754-2008. During its 23 years, it was the most widely used format for floating-point computation. It was implemented in software, in the form of floating-point libraries, and in hardware, in the instructions of many CPUs and FPUs. The first integrated circuit to implement the draft of what was to become IEEE 754-1985 was the Intel 8087.
IEEE 754-1985 represents numbers in binary, providing definitions for four levels of precision, of which the two most commonly used are:
Level
Width
Range at full precision
Precision[a]
Single precision 32 bits ±1.18×10−38 to ±3.4×1038 Approximately 7 decimal digits
Double precision 64 bits ±2.23×10−308 to ±1.80×10308 Approximately 16 decimal digits
The standard also defines representations for positive and negative infinity, a "negative zero", five exceptions to handle invalid results like division by zero, special values called NaNs for representing those exceptions, denormal numbers to represent numbers smaller than shown above, and four rounding modes.
Contents
[hide]
1 Representation of numbers
1.1 Zero
1.2 Denormalized numbers
2 Representation of non-numbers
2.1 Positive and negative infinity
2.2 NaN
3 Range and precision
3.1 Single precision
3.2 Double precision
3.3 Extended formats
4 Examples
5 Comparing floating-point numbers
6 Rounding floating-point numbers
7 Extending the real numbers
8 Functions and predicates
8.1 Standard operations
8.2 Recommended functions and predicates
9 History
10 See also
11 Notes
12 References
13 Further reading
14 External links
Representation of numbers[edit]
The number 0.15625 represented as a single-precision IEEE 754-1985 floating-point number. See text for explanation.
The three fields in a 64bit IEEE 754 float
Floating-point numbers in IEEE 754 format consist of three fields: a sign bit, a biased exponent, and a fraction. The following example illustrates the meaning of each.
The decimal number 0.1562510 represented in binary is 0.001012 (that is, 1/8 + 1/32). (Subscripts indicate the number base.) Analogous to scientific notation, where numbers are written to have a single non-zero digit to the left of the decimal point, we rewrite this number so it has a single 1 bit to the left of the "binary point". We simply multiply by the appropriate power of 2 to compensate for shifting the bits left by three positions:
0.00101 2 = 1.01 2 × 2 − 3 {\displaystyle 0.00101_{2}=1.01_{2}\times 2^{-3}}
Now we can read off the fraction and the exponent: the fraction is .012 and the exponent is −3.
As illustrated in the pictures, the three fields in the IEEE 754 representation of this number are:
sign = 0, because the number is positive. (1 indicates negative.)
biased exponent = −3 + the "bias". In single precision, the bias is 127, so in this example the biased exponent is 124; in double precision, the bias is 1023, so the biased exponent in this example is 1020.
fraction = .01000…2.
IEEE 754 adds a bias to the exponent so that numbers can in many cases be compared conveniently by the same hardware that compares signed 2's-complement integers. Using a biased exponent, the lesser of two positive floating-point numbers will come out "less than" the greater following the same ordering as for sign and magnitude integers. If two floating-point numbers have different signs, the sign-and-magnitude comparison also works with biased exponents. However, if both biased-exponent floating-point numbers are negative, then the ordering must be reversed. If the exponent were represented as, say, a 2's-complement number, comparison to see which of two numbers is greater would not be as convenient.
The leading 1 bit is omitted since all numbers except zero start with a leading 1; the leading 1 is implicit and doesn't actually need to be stored which gives an extra bit of precision for "free."
Zero[edit]
The number zero is represented specially:
sign = 0 for positive zero, 1 for negative zero.
biased exponent = 0.
fraction = 0.
Denormalized numbers[edit]
The number representations described above are called normalized, meaning that the implicit leading binary digit is a 1. To reduce the loss of precision when an underflow occurs, IEEE 754 includes the ability to represent fractions smaller than are possible in the normalized representation, by making the implicit leading digit a 0. Such numbers are called denormal. They don't include as many significant digits as a normalized number, but they enable a gradual loss of precision when the result of an arithmetic operation is not exactly zero but is too close to zero to be represented by a normalized number.
A denormal number is represented with a biased exponent of all 0 bits, which represents an exponent of −126 in single precision (not −127), or −1022 in double precision (not −1023).[1] In contrast, the smallest biased exponent representing a normal number is 1 (see examples below).
Representation of non-numbers[edit]
The biased-exponent field is filled with all 1 bits to indicate either infinity or an invalid result of a computation.
Positive and negative infinity[edit]
Positive and negative infinity are represented thus:
sign = 0 for positive infinity, 1 for negative infinity.
biased exponent = all 1 bits.
fraction = all 0 bits.
NaN[edit]
Some operations of floating-point arithmetic are invalid, such as taking the square root of a negative number. The act of reaching an invalid result is called a floating-point exception. An exceptional result is represented by a special code called a NaN, for "Not a Number". All NaNs in IEEE 754-1985 have this format:
sign = either 0 or 1.
biased exponent = all 1 bits.
fraction = anything except all 0 bits (since all 0 bits represents infinity).
Range and precision[edit]
Relative precision of single (binary32) and double precision (binary64) numbers, compared with decimal representations using a fixed number of significant digits. Relative precision is defined here as ulp(x)/x, where ulp(x) is the unit in the last place in the representation of x, i.e. the gap between x and the next representable number.
Precision is defined as the minimum difference between two successive mantissa representations; thus it is a function only in the mantissa; while the gap is defined as the difference between two successive numbers.[2]
Single precision[edit]
Single-precision numbers occupy 32 bits. In single precision:
The positive and negative numbers closest to zero (represented by the denormalized value with all 0s in the exponent field and the binary value 1 in the fraction field) are
±2−149 ≈ ±1.40130×10−45
The positive and negative normalized numbers closest to zero (represented with the binary value 1 in the exponent field and 0 in the fraction field) are
±2−126 ≈ ±1.17549×10−38
The finite positive and finite negative numbers furthest from zero (represented by the value with 254 in the exponent field and all 1s in the fraction field) are
±(1−2−24) × 2128[3] ≈ ±3.40282×1038
Some example range and gap values for given exponents in single precision:
Actual Exponent (unbiased)
Exp (biased)
Minimum
Maximum
Gap
0 127 1 ≈ 1.999999880791 ≈ 1.19209e-7
1 128 2 ≈ 3.999999761581 ≈ 2.38419e-7
2 129 4 ≈ 7.999999523163 ≈ 4.76837e-7
10 137 1024 ≈ 2047.999877930 ≈ 1.22070e-4
11 138 2048 ≈ 4095.999755859 ≈ 2.44141e-4
23 150 8388608 16777215 1
24 151 16777216 33554430 2
127 254 ≈ 1.70141e38 ≈ 3.40282e38 ≈ 2.02824e31
As an example, 16,777,217 can not be encoded as a 32-bit float as it will be rounded to 16,777,216. This shows why floating point arithmetic is unsuitable for accounting software. However, all integers within the representable range that are a power of 2 can be stored in a 32-bit float without rounding.
Double precision[edit]
Double-precision numbers occupy 64 bits. In double precision:
The positive and negative numbers closest to zero (represented by the denormalized value with all 0s in the Exp field and the binary value 1 in the Fraction field) are
±2−1074 ≈ ±4.94066×10−324
The positive and negative normalized numbers closest to zero (represented with the binary value 1 in the Exp field and 0 in the fraction field) are
±2−1022 ≈ ±2.22507×10−308
The finite positive and finite negative numbers furthest from zero (represented by the value with 2046 in the Exp field and all 1s in the fraction field) are
±(1−2−53) × 21024[3] ≈ ±1.79769×10308
Some example range and gap values for given exponents in double precision:
Actual Exponent (unbiased)
Exp (biased)
Minimum
Maximum
Gap
0 1023 1 ≈ 1.999999999999999777955 ≈ 2.22045e-16
1 1024 2 ≈ 3.999999999999999555911 ≈ 4.44089e-16
2 1025 4 ≈ 7.999999999999999111822 ≈ 8.88178e-16
10 1033 1024 ≈ 2047.999999999999772626 ≈ 2.27374e-13
11 1034 2048 ≈ 4095.999999999999545253 ≈ 4.54747e-13
52 1075 4503599627370496 9007199254740991 1
53 1076 9007199254740992 18014398509481982 2
1023 2046 ≈ 8.98847e307 ≈ 1.79769e308 ≈ 1.99584e292
Extended formats[edit]
The standard also recommends extended format(s) to be used to perform internal computations at a higher precision than that required for the final result, to minimise round-off errors: the standard only specifies minimum precision and exponent requirements for such formats. The x87 80-bit extended format is the most commonly implemented extended format that meets these requirements.
Examples[edit]
Here are some examples of single-precision IEEE 754 representations:
Type
Sign
Actual Exponent
Exp (biased)
Exponent field
Significand (fraction field)
Value
Zero 0 −127 0 0000 0000 000 0000 0000 0000 0000 0000 0.0
Negative zero 1 −127 0 0000 0000 000 0000 0000 0000 0000 0000 −0.0
One 0 0 127 0111 1111 000 0000 0000 0000 0000 0000 1.0
Minus One 1 0 127 0111 1111 000 0000 0000 0000 0000 0000 −1.0
Smallest denormalized number * −126 0 0000 0000 000 0000 0000 0000 0000 0001 ±2−23 × 2−126 = ±2−149 ≈ ±1.4×10−45
"Middle" denormalized number * −126 0 0000 0000 100 0000 0000 0000 0000 0000 ±2−1 × 2−126 = ±2−127 ≈ ±5.88×10−39
Largest denormalized number * −126 0 0000 0000 111 1111 1111 1111 1111 1111 ±(1−2−23) × 2−126 ≈ ±1.18×10−38
Smallest normalized number * −126 1 0000 0001 000 0000 0000 0000 0000 0000 ±2−126 ≈ ±1.18×10−38
Largest normalized number * 127 254 1111 1110 111 1111 1111 1111 1111 1111 ±(2−2−23) × 2127 ≈ ±3.4×1038
Positive infinity 0 128 255 1111 1111 000 0000 0000 0000 0000 0000 +∞
Negative infinity 1 128 255 1111 1111 000 0000 0000 0000 0000 0000 −∞
Not a number * 128 255 1111 1111 non zero NaN
* Sign bit can be either 0 or 1 .
Comparing floating-point numbers[edit]
Every possible bit combination is either a NaN or a number with a unique value in the affinely extended real number system with its associated order, except for the two bit combinations negative zero and positive zero, which sometimes require special attention (see below). The binary representation has the special property that, excluding NaNs, any two numbers can be compared as sign and magnitude integers (endianness issues apply). When comparing as 2's-complement integers: If the sign bits differ, the negative number precedes the positive number, so 2's complement gives the correct result (except that negative zero and positive zero should be considered equal). If both values are positive, the 2's complement comparison again gives the correct result. Otherwise (two negative numbers), the correct FP ordering is the opposite of the 2's complement ordering.
Rounding errors inherent in floating point calculations often make comparison of results for exact equality not useful. Choosing an acceptable range is a complex topic.
Although negative zero and positive zero are generally considered equal for comparison purposes, some programming language relational operators and similar constructs might or do treat them as distinct. According to the Java Language Specification,[4] comparison and equality operators treat them as equal, but Math.min() and Math.max() distinguish them (officially starting with Java version 1.1 but actually with 1.1.1), as do the comparison methods equals(), compareTo() and even compare() of classes Float and Double.
Rounding floating-point numbers[edit]
The IEEE standard has four different rounding modes; the first is the default; the others are called directed roundings.
Round to Nearest – rounds to the nearest value; if the number falls midway it is rounded to the nearest value with an even (zero) least significant bit, which occurs 50% of the time (in IEEE 754-2008 this mode is called roundTiesToEven to distinguish it from another round-to-nearest mode)
Round toward 0 – directed rounding towards zero
Round toward +∞ – directed rounding towards positive infinity
Round toward −∞ – directed rounding towards negative infinity.
Extending the real numbers[edit]
The IEEE standard employs (and extends) the affinely extended real number system, with separate positive and negative infinities. During drafting, there was a proposal for the standard to incorporate the projectively extended real number system, with a single unsigned infinity, by providing programmers with a mode selection option. In the interest of reducing the complexity of the final standard, the projective mode was dropped, however. The Intel 8087 and Intel 80287 floating point co-processors both support this projective mode.[5][6][7]
Functions and predicates[edit]
Standard operations[edit]
The following functions must be provided:
Add, subtract, multiply, divide
Square root
Floating point remainder. This is not like a normal modulo operation, it can be negative for two positive numbers. It returns the exact value of x–(round(x/y)·y).
Round to nearest integer. For undirected rounding when halfway between two integers the even integer is chosen.
Comparison operations. Besides the more obvious results, IEEE 754 defines that −∞ = −∞, +∞ = +∞ and x ≠ NaN for any x (including NaN).
Recommended functions and predicates[edit]
copysign(x,y) returns x with the sign of y, so abs(x) equals copysign(x,1.0). This is one of the few operations which operates on a NaN in a way resembling arithmetic. The function copysign is new in the C99 standard.
−x returns x with the sign reversed. This is different from 0−x in some cases, notably when x is 0. So −(0) is −0, but the sign of 0−0 depends on the rounding mode.
scalb(y, N)
logb(x)
finite(x) a predicate for "x is a finite value", equivalent to −Inf < x < Inf
isnan(x) a predicate for "x is a NaN", equivalent to "x ≠ x"
x <> y which turns out to have different exception behavior than NOT(x = y).
unordered(x, y) is true when "x is unordered with y", i.e., either x or y is a NaN.
class(x)
nextafter(x,y) returns the next representable value from x in the direction towards y
History[edit]
In 1976 Intel began planning to produce a floating point coprocessor. John Palmer, the manager of the effort, persuaded them that they should try to develop a standard for all their floating point operations. William Kahan was hired as a consultant; he had helped improve the accuracy of Hewlett-Packard's calculators. Kahan initially recommended that the floating point base be decimal[8] but the hardware design of the coprocessor was too far along to make that change.
The work within Intel worried other vendors, who set up a standardization effort to ensure a 'level playing field'. Kahan attended the second IEEE 754 standards working group meeting, held in November 1977. Here, he received permission from Intel to put forward a draft proposal based on the standard arithmetic part of their design for a coprocessor. The arguments over gradual underflow lasted until 1981 when an expert hired by DEC to assess it sided against the dissenters.
Even before it was approved, the draft standard had been implemented by a number of manufacturers.[9][10] The Intel 8087, which was announced in 1980, was the first chip to implement the draft standard.
See also[edit]
−0 (negative zero)
Intel 8087
minifloat for simple examples of properties of IEEE 754 floating point numbers
Fixed-point arithmetic
Notes[edit]
Jump up ^ Precision: The number of decimal digits precision is calculated via number_of_mantissa_bits * Log10(2). Thus ~7.2 and ~15.9 for single and double precision respectively.
References[edit]
Jump up ^ Hennessy. Computer Organization and Design. Morgan Kaufmann. p. 270.
Jump up ^ Hossam A. H. Fahmy; Shlomo Waser; Michael J. Flynn, Computer Arithmetic (PDF), archived from the original (PDF) on 2010-10-08, retrieved 2011-01-02
^ Jump up to: a b William Kahan. "Lecture Notes on the Status of IEEE 754" (PDF). October 1, 1997 3:36 am. Elect. Eng. & Computer Science University of California. Retrieved 2007-04-12.
Jump up ^ "Java Language and Virtual Machine Specifications". Java Documentation.
Jump up ^ John R. Hauser (March 1996). "Handling Floating-Point Exceptions in Numeric Programs" (PDF). ACM Transactions on Programming Languages and Systems. 18 (2): 139–174. doi:10.1145/227699.227701.
Jump up ^ David Stevenson (March 1981). "IEEE Task P754: A proposed standard for binary floating-point arithmetic". IEEE Computer. 14 (3): 51–62.
Jump up ^ William Kahan and John Palmer (1979). "On a proposed floating-point standard". SIGNUM Newsletter. 14 (Special): 13–21. doi:10.1145/1057520.1057522.
Jump up ^ W. Kahan 2003, pers. comm. to Mike Cowlishaw and others after an IEEE 754 meeting[unreliable source?]
Jump up ^ Charles Severance (20 February 1998). "An Interview with the Old Man of Floating-Point".
Jump up ^ Charles Severance. "History of IEEE Floating-Point Format". Connexions.
Further reading[edit]
Charles Severance (March 1998). "IEEE 754: An Interview with William Kahan" (PDF). IEEE Computer. 31 (3): 114–115. doi:10.1109/MC.1998.660194. Retrieved 2008-04-28.
David Goldberg (March 1991). "What Every Computer Scientist Should Know About Floating-Point Arithmetic" (PDF). ACM Computing Surveys. 23 (1): 5–48. doi:10.1145/103162.103163. Retrieved 2008-04-28.
Chris Hecker (February 1996). "Let's Get To The (Floating) Point" (PDF). Game Developer Magazine: 19–24. ISSN 1073-922X.
David Monniaux (May 2008). "The pitfalls of verifying floating-point computations". ACM Transactions on Programming Languages and Systems. 30 (3): article #12. doi:10.1145/1353445.1353446. ISSN 0164-0925. : A compendium of non-intuitive behaviours of floating-point on popular architectures, with implications for program verification and testing.
External links[edit]
Comparing floats
Coprocessor.info: x87 FPU pictures, development and manufacturer information
IEEE 854-1987 — History and minutes
IEEE754 (Single and Double precision) Online Converter
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