In computer science and numerical analysis, unit in the last place or unit of least precision (ulp) is the spacing between two consecutive floating-point numbers, i.e., the value the least significant digit (rightmost digit) represents if it is 1. It is used as a measure of accuracy in numeric calculations.^[1]

Definition

One definition is: In radix ${\displaystyle b}$ with precision ${\displaystyle p}$, if ${\displaystyle b^{e}\leq |x|<b^{e+1}}$, then ${\displaystyle \operatorname {ulp} (x)=b^{\max\{e,\,e_{\min }\}-p+1}}$.^[2]

Another definition, suggested by John Harrison, is slightly different: ${\displaystyle \operatorname {ulp} (x)}$ is the distance between the two closest straddling floating-point numbers ${\displaystyle a}$ and ${\displaystyle b}$ (i.e., those with ${\displaystyle a\leq x\leq b}$ and ${\displaystyle a\neq b}$), assuming that the exponent range is not upper-bounded.^[3][4] These definitions differ only at signed powers of the radix.^[2]

The IEEE 754 specification—followed by all modern floating-point hardware—requires that the result of an elementary arithmetic operation (addition, subtraction, multiplication, division, and square root since 1985, and FMA since 2008) be correctly rounded, which implies that in rounding to nearest, the rounded result is within 0.5 ulp of the mathematically exact result, using John Harrison's definition; conversely, this property implies that the distance between the rounded result and the mathematically exact result is minimized (but for the halfway cases, it is satisfied by two consecutive floating-point numbers). Reputable numeric libraries compute the basic transcendental functions to between 0.5 and about 1 ulp. Only a few libraries compute them within 0.5 ulp, this problem being complex due to the Table-maker's dilemma.^[5]

Examples

> until (\x -> x == x+1) (+1) 0 :: Float
1.6777216e7
> it-1
1.6777215e7
> it+1
1.6777216e7

// π with 20 decimal digits
BigDecimal π = new BigDecimal("3.14159265358979323846");

// truncate to a double floating point
double p0 = π.doubleValue();
// -> 3.141592653589793  (hex: 0x1.921fb54442d18p1)

// p0 is smaller than π, so find next number representable as double
double p1 = Math.nextUp(p0);
// -> 3.1415926535897936 (hex: 0x1.921fb54442d19p1)

// ulp(π) is the difference between p1 and p0
BigDecimal ulp = new BigDecimal(p1).subtract(new BigDecimal(p0));
// -> 4.44089209850062616169452667236328125E-16
// (this is precisely 2**(-51))

// same result when using the standard library function
double ulpMath = Math.ulp(p0);
// -> 4.440892098500626E-16 (hex: 0x1.0p-51)

>>> x = 1.0
>>> p = 0
>>> while x != x + 1:
...   x = x * 2
...   p = p + 1
... 
>>> x
9007199254740992.0
>>> p
53
>>> x + 2 + 1
9007199254740996.0

Language support

The Boost C++ libraries provides the functions boost::math::float_next, boost::math::float_prior, boost::math::nextafter and boost::math::float_advance to obtain nearby (and distant) floating-point values,^[6] and boost::math::float_distance(a, b) to calculate the floating-point distance between two doubles.^[7]

The C language library provides functions to calculate the next floating-point number in some given direction: nextafterf and nexttowardf for float, nextafter and nexttoward for double, nextafterl and nexttowardl for long double, declared in <math.h>. It also provides the macros FLT_EPSILON, DBL_EPSILON, LDBL_EPSILON, which represent the positive difference between 1.0 and the next greater representable number in the corresponding type (i.e. the ulp of one).^[8]

The Java standard library provides the functions Math.ulp(double) and Math.ulp(float). They were introduced with Java 1.5.

The Swift standard library provides access to the next floating-point number in some given direction via the instance properties nextDown and nextUp. It also provides the instance property ulp and the type property ulpOfOne (which corresponds to C macros like FLT_EPSILON^[9]) for Swift's floating-point types.^[10]

See also

• IEEE 754
• ISO/IEC 10967, part 1 requires an ulp function
• Least significant bit (LSB)
• Machine epsilon
• Rounding error

References

Bibliography

Look up ulp in Wiktionary, the free dictionary.

• Goldberg, David (1991–03). "Rounding Error" in "What Every Computer Scientist Should Know About Floating-Point Arithmetic". Computing Surveys, ACM, March 1991. Retrieved from http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html#689.
• Muller, Jean-Michel (2010). Handbook of floating-point arithmetic. Boston: Birkhäuser. pp. 32–37. ISBN 978-0-8176-4704-9.