Unit in the last place

The most common definition is: In radix b with precision p, if b^e \le |x| , then {{nowrap|\operatorname{ulp}(x) = b^{\max \{ e, \, e_\min \} - p + 1},}} where e_\min is the minimal exponent of the normal numbers. In particular, \operatorname{ulp}(x) = b^{e - p + 1} for normal numbers, and \operatorname{ulp}(x) = b^{e_\min - p + 1} for subnormals. Another definition, suggested by John Harrison, is slightly different: \operatorname{ulp}(x) is the distance between the two closest straddling floating-point numbers a and b (i.e., satisfying a \le x \le b and a \neq b), assuming that the exponent range is not upper-bounded. These definitions differ only at signed powers of the radix. Since the 2010s, advances in floating-point mathematics have allowed correctly rounded functions to be almost as fast in average as these earlier, less accurate functions. A correctly rounded function would also be fully reproducible. which theoretically would only produce one incorrect rounding out of 1000 random floating-point inputs. ==Examples==

Example 1 Let x be a positive floating-point number and assume that the active rounding mode is round to nearest, ties to even, denoted \operatorname{RN}. If \operatorname{ulp}(x) \le 1, then \operatorname{RN} (x + 1) > x. Otherwise, \operatorname{RN} (x + 1) = x or \operatorname{RN} (x + 1) = x + \operatorname{ulp}(x), depending on the value of the least significant digit and the exponent of x. This is demonstrated in the following Haskell code typed at an interactive prompt: > until (\x -> x == x+1) (+1) 0 :: Float 1.6777216e7 > it-1 1.6777215e7 > it+1 1.6777216e7 Here we start with 0 in single precision (binary32) and repeatedly add 1 until the operation does not change the value. Since the significand for a single-precision number contains 24 bits, the first integer that is not exactly representable is 224+1, and this value rounds to 224 in round to nearest, ties to even. Thus the result is equal to 224. Example 2 The following example in Java approximates Pi| as a floating-point value by finding the two double values bracketing \pi: p_0 . // 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) Then \operatorname{ulp}(\pi) is determined as \operatorname{ulp}(\pi) = p_1 - p_0. // 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) Example 3 Another example, in Python, also typed at an interactive prompt, is: >>> 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 In this case, we start with x = 1 and repeatedly double it until x = x + 1. Similarly to Example 1, the result is 253 because the double-precision floating-point format uses a 53-bit significand. ==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, and boost::math::float_distance(a, b) to calculate the floating-point distance between two doubles. 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 . 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). The Go standard library provides the functions math.Nextafter (for 64 bit floats) and math.Nextafter32 (for 32 bit floats) both of which return the next representable floating-point value towards another provided floating-point value. The Java standard library provides the functions and . 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) for Swift's floating-point types. ==See also==