Python math.ulp() Function
Python math ModuleIn floating-point calculations, understandingthe precision limitations of floating-point numbersis very important. ULP (Unit in the Last Place) is the basic unit for measuring the precision of floating-point numbers.
math.ulp()is a function introduced in Python 3.9, used to return thesmallest significant floating-point unit(ULP), that is, the distance from that number to the next representable floating-point number.
Word Definition： ulpis the abbreviation of "Unit in the Last Place", meaning "the unit of the last place".
Basic Syntax and Parameters
Syntax Format
import math math.ulp(x)
Parameter Description
- x: floating-point number
Return Value
Returns the ULP value of x, that is, the difference between x and the next representable floating-point number
Examples
Example 1: Basic Usage
Example
import math
print("ULP of 1.0:", math.ulp(1.0))
print("ULP of 2.0:", math.ulp(2.0))
print("ULP of 100.0:", math.ulp(100.0))
print("ULP of 0.0:", math.ulp(0.0))
print("Smallest positive number:", math.ulp(float.min)
print("ULP of 1.0:", math.ulp(1.0))
print("ULP of 2.0:", math.ulp(2.0))
print("ULP of 100.0:", math.ulp(100.0))
print("ULP of 0.0:", math.ulp(0.0))
print("Smallest positive number:", math.ulp(float.min)
Output:
1.0 的 ULP: 2.220446049250313e-16 2.0 的 ULP: 4.440892098500626e-16 100.0 的 ULP: 2.8421709430404007e-14 0.0 的 ULP: 5e-324
Example 2: ULP for Different Values
Example
import math
values = [0.1, 0.5, 1.0, 1.5, 2.0, 10.0, 100.0, 1000.0]
print("ULP for different values:")
for x in values:
print(f" ULP({x}) = {math.ulp(x)}")
values = [0.1, 0.5, 1.0, 1.5, 2.0, 10.0, 100.0, 1000.0]
print("ULP for different values:")
for x in values:
print(f" ULP({x}) = {math.ulp(x)}")
Output:
不同数值的 ULP: ULP(0.1) = 1.3877787807814457e-17 ULP(0.5) = 1.3877787807814457e-16 ULP(1.0) = 2.220446049250313e-16 ULP(1.5) = 2.220446049250313e-16 ULP(2.0) = 4.440892098500626e-16 U浮点 10.0: 2.220446049250313e-15 ULP(100.0) = 2.8421709430404007e-14 ULP(1000.0) = 2.2737367544323206e-13
Example 3: Special Values
Example
import math
print("Infinity:", math.ulp(math.inf))
print("Largest finite number:", math.ulp(sys.float_info.max))
print("Negative number:", math.ulp(-1.0))
print("Subnormal number:", math.ulp(1e-310))
print("Infinity:", math.ulp(math.inf))
print("Largest finite number:", math.ulp(sys.float_info.max))
print("Negative number:", math.ulp(-1.0))
print("Subnormal number:", math.ulp(1e-310))
Output:
无穷大: inf 最大有限数: inf 负数: 2.220446049250313e-16 次正规数: 5e-324
Example 4: Floating-Point Precision Detection
Example
import math
def check_precision(x):
"""Detect floating-point precision"""
ulp = math.ulp(x)
relative_error = ulp / x
print(f"x = {x}")
print(f" ULP = {ulp}")
print(f" Relative error ≈ {relative_error:.2e}")
print(f" Significant digits ≈ {-math.log2(relative_error):.1f}")
print()
check_precision(1.0)
check_precision(1000000.0)
def check_precision(x):
"""Detect floating-point precision"""
ulp = math.ulp(x)
relative_error = ulp / x
print(f"x = {x}")
print(f" ULP = {ulp}")
print(f" Relative error ≈ {relative_error:.2e}")
print(f" Significant digits ≈ {-math.log2(relative_error):.1f}")
print()
check_precision(1.0)
check_precision(1000000.0)
Output:
x = 1.0 ULP = 2.220446049250313e-16 相对误差 ≈ 2.22e-16 有效位数 ≈ 52.0 x = 1000000.0 ULP = 1.907344663e-13 相对误差 ≈ 1.91e-13 有效位数 ≈ 52.0
Example 5: Combining with nextafter
Example
import math
# Verify: nextafter(x, inf) - x = ULP(x)
x = 1.0
next_val = math.nextafter(x, math.inf)
ulp_calc = math.ulp(x)
direct_diff = next_val - x
print(f"x = {x}")
print(f"nextafter(x, inf) - x = {direct_diff}")
print(f"math.ulp(x) = {ulp_calc}")
print(f"Equal: {direct_diff == ulp_calc}")
# Verify: nextafter(x, inf) - x = ULP(x)
x = 1.0
next_val = math.nextafter(x, math.inf)
ulp_calc = math.ulp(x)
direct_diff = next_val - x
print(f"x = {x}")
print(f"nextafter(x, inf) - x = {direct_diff}")
print(f"math.ulp(x) = {ulp_calc}")
print(f"Equal: {direct_diff == ulp_calc}")
Output:
x = 1.0 nextafter(x, inf) - x = 2.220446049250313e-16 math.ulp(x) = 2.220 double 000001e-16 两者相等: True
Note: math.ulp(x) is equivalent to |nextafter(x, inf) - x|
Application Scenarios
- Error analysis of numerical algorithms
- Floating-point precision detection
- Tolerance setting in scientific computing
- Determining an appropriate epsilon when comparing floating-point numbers
Notes
- This function is only available in Python 3.9+
- The ULP value increases as the number increases
- The ULP of inf is also inf
Other Extensions