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Some are applied by hand, while others are employed by digital circuit designs and software. Division algorithms fall into two main categories: Slow division algorithms produce one digit of the final quotient per iteration. Examples of slow division include restoring , non-performing restoring, non-restoring , and SRT division. Fast division methods start with a close approximation to the final quotient and produce twice as many digits of the final quotient on each iteration.
Newton—Raphson and Goldschmidt fall into this category. The simplest division algorithm, historically incorporated into a greatest common divisor algorithm presented in Euclid's Elements , Book VII, Proposition 1, finds the remainder given two positive integers using only subtractions and comparisons:. The proof that the quotient and remainder exist and are unique described at Euclidean division gives rise to a complete division algorithm using additions, subtractions, and comparisons:.
It is useful if Q is known to be small being an output-sensitive algorithm , and can serve as an executable specification. Long division is the standard algorithm used for pen-and-paper division of multidigit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor at each stage; the multiples become the digits of the quotient, and the final difference is the remainder.
When used with a binary radix, it forms the basis for the integer division unsigned with remainder algorithm below. Short division is an abbreviated form of long division suitable for one-digit divisors. Chunking also known as the partial quotients method or the hangman method is a less-efficient form of long division which may be easier to understand. The following algorithm, the binary version of the famous long division , will divide N by D , placing the quotient in Q and the remainder in R.
In the following code, all values are treated as unsigned integers. Restoring division operates on fixed-point fractional numbers and depends on the following assumptions: The above restoring division algorithm can avoid the restoring step by saving the shifted value 2 P before the subtraction in an additional register T i. This lets it be executed faster. The basic algorithm for binary radix 2 non-restoring division of non-negative numbers is:. This form needs to be converted to binary to form the final quotient.
To convert to a positive remainder, do a single restoring step after Q is converted from non-standard form to standard form:. As with restoring division, the low-order bits of P are used up at the same rate as bits of the quotient Q are produced, and it is common to use a single shift register for both. Named for its creators Sweeney, Robertson, and Tocher , SRT division is a popular method for division in many microprocessor implementations. The Intel Pentium processor's infamous floating-point division bug was caused by an incorrectly coded lookup table.
Five of the entries had been mistakenly omitted. This squaring of the error at each iteration step — the so-called quadratic convergence of Newton—Raphson's method — has the effect that the number of correct digits in the result roughly doubles for every iteration , a property that becomes extremely valuable when the numbers involved have many digits e. Apply a bit-shift to the divisor D to scale it so that 0.
The same bit-shift should be applied to the numerator N so that the quotient does not change. Then one could use a linear approximation in the form.
To minimize the maximum of the relative error of this approximation on interval [ 0. The coefficients of the linear approximation are determined as follows.
Using this approximation, the relative error of the initial value is less than. It is possible to generate a polynomial fit of degree larger than 1, computing the coefficients using the Remez algorithm. The trade-off is that the initial guess requires more computational cycles but hopefully in exchange for fewer iterations of Newton—Raphson. Since for this method the convergence is exactly quadratic, it follows that. This evaluates to 3 for IEEE single precision and 4 for both double precision and double extended formats.
For example, for a double-precision floating-point division, this method uses 10 multiplies, 9 adds, and 2 shifts. Goldschmidt after Robert Elliott Goldschmidt  division uses an iterative process of repeatedly multiplying both the dividend and divisor by a common factor F i , chosen such that the divisor converges to 1. This causes the dividend to converge to the sought quotient Q:. The Goldschmidt method can be used with factors that allow simplifications by the binomial theorem.
Methods designed for hardware implementation generally do not scale to integers with thousands or millions of decimal digits; these frequently occur, for example, in modular reductions in cryptography. The result is that the computational complexity of the division is of the same order up to a multiplicative constant as that of the multiplication.
Examples include reduction to multiplication by Newton's method as described above ,  as well as the slightly faster Barrett reduction and Montgomery reduction algorithms. The division by a constant D is equivalent to the multiplication by its reciprocal.
Consequently, if Y were a power of two the division step would reduce to a fast right bit shift. However, unless D itself is a power of two, there is no X and Y that satisfies the conditions above. In some cases, division by a constant can be accomplished in even less time by converting the "multiply by a constant" into a series of shifts and adds or subtracts.
Round-off error can be introduced by division operations due to limited precision. From Wikipedia, the free encyclopedia. This article is about algorithms for division. For the theorem proving the existence of a unique quotient and remainder, see Euclidean division. This section needs expansion. You can help by adding to it. Architectures, Models, and Implementations; Stanford University; Retrieved 22 October Fast Division of Large Integers: Royal Institute of Technology. Archived from the original PDF on 8 July Faster Unsigned Division by Constants".
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