Sterbenz lemma

In floating-point arithmetic, the Sterbenz lemma or Sterbenz' lemma^[1] is a theorem giving conditions under which floating-point differences are computed exactly. It is named after Pat H. Sterbenz, who published it as Theorem 4.3.1 in his 1974 book Floating-Point Computation.^[2]

The Sterbenz lemma states that for a floating-point number system with subnormal numbers, such as IEEE 754 binary64, if ${\displaystyle x}$ and ${\displaystyle y}$ are floating-point numbers such that

then ${\displaystyle x-y}$ is also a floating-point number. Thus, the floating-point subtraction ${\displaystyle x\ominus y=\operatorname {fl} (x-y)=x-y}$ is computed exactly.

Relation to catastrophic cancellation[]

The Sterbenz lemma may be contrasted with the phenomenon of catastrophic cancellation:

• The Sterbenz lemma asserts that if ${\displaystyle x}$ and ${\displaystyle y}$ are sufficiently close floating-point numbers then their difference ${\displaystyle x-y}$ is computed exactly by floating-point arithmetic ${\displaystyle x\ominus y=\operatorname {fl} (x-y)}$, with no rounding needed.
• The phenomenon of catastrophic cancellation is that if ${\displaystyle {\tilde {x}}}$ and ${\displaystyle {\tilde {y}}}$ are approximations to true numbers ${\displaystyle x}$ and ${\displaystyle y}$—whether the approximations arise from prior rounding error or from series truncation or from physical uncertainty or anything else—the error of the difference ${\displaystyle {\tilde {x}}-{\tilde {y}}}$ from the desired difference ${\displaystyle x-y}$ is inversely proportional to ${\displaystyle x-y}$. Thus, the closer ${\displaystyle x}$ and ${\displaystyle y}$ are, the worse ${\displaystyle {\tilde {x}}-{\tilde {y}}}$ may be as an approximation to ${\displaystyle x-y}$, even if the subtraction itself is computed exactly.

In other words, the Sterbenz lemma shows that subtracting nearby floating-point numbers is exact, but if the numbers you have are approximations then even their exact difference may be far off from the difference of numbers you wanted to subtract.

Use in numerical analysis[]

The Sterbenz lemma is instrumental in proving theorems on error bounds in numerical analysis of floating-point algorithms. For example, Heron's formula

for the area of triangle with side lengths ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$, where ${\displaystyle s=(a+b+c)/2}$ is the semi-perimeter, may give poor accuracy for long narrow triangles if evaluated directly in floating-point arithmetic. However, the alternative formula can be proven, with the help of the Sterbenz lemma, to have low forward error for all inputs.^[3][4][5]

References[]