When a colleague sent Barclay a strain of SARS-CoV-2 in culture that had spontaneously lost the furin cleavage site, her team found that ferrets infected with this strain shed viral particles in lower amounts than did those infected with the pandemic strain, and did not transmit the infection to nearby animals 9. Furin is suspected to cut the site at some point during virion assembly, or just before release.
The timing might explain why the virus exits through the Golgi or lysosomes, says Tom Gallagher, a virologist at Loyola University Chicago in Illinois. By snipping the bond between the S1 and S2 subunits, the furin cut loosens up virion spike proteins so that during cell entry they respond to a second cut by TMPRSS2, which exposes the hydrophobic area that rapidly buries itself in a host-cell membrane, says Gallagher.
Two coronavirus variants, Alpha and Delta, have altered furin cleavage sites. In the Alpha variant, the initial proline amino acid is changed to a histidine PH ; in the Delta variant, it is changed to an arginine PR. Both changes make the sequence less acidic, and the more basic the string of amino acids, the more effectively furin recognizes and cuts it, says Barclay.
More furin cuts mean more spike proteins primed to enter human cells. It is not easy to keep pace with the quickly mutating virus.
Most mutations so far are associated with how effectively the virus spreads, not with how much the virus damages the host, experts agree. Casalino, L. ACS Cent. PubMed Article Google Scholar. Ke, Z. Nature , — Science , — Nguyen, H. B , — Shang, J. Gobeil, S. Khateeb, J. Care 25 , Hoffmann, M. Cell , — Peacock, T. Nature Microbiol. Wang, M. Cell Res. Gunst, J. EClinicalMedicine 35 , Article Google Scholar.
Finkel, Y. Schubert, K. Nature Struct. Thoms, M. Then exp 1. Should this be rounded to 5. If exp 1. And then 5. Since exp is transcendental, this could go on arbitrarily long before distinguishing whether exp 1. Thus it is not practical to specify that the precision of transcendental functions be the same as if they were computed to infinite precision and then rounded.
Another approach would be to specify transcendental functions algorithmically. But there does not appear to be a single algorithm that works well across all hardware architectures.
Rational approximation, CORDIC, 16 and large tables are three different techniques that are used for computing transcendentals on contemporary machines. Each is appropriate for a different class of hardware, and at present no single algorithm works acceptably over the wide range of current hardware.
On some floating-point hardware every bit pattern represents a valid floating-point number. On the other hand, the VAX TM reserves some bit patterns to represent special numbers called reserved operands. Without any special quantities, there is no good way to handle exceptional situations like taking the square root of a negative number, other than aborting computation. Since every bit pattern represents a valid number, the return value of square root must be some floating-point number.
However, there are examples where it makes sense for a computation to continue in such a situation. Consider a subroutine that finds the zeros of a function f , say zero f. Traditionally, zero finders require the user to input an interval [ a , b ] on which the function is defined and over which the zero finder will search.
That is, the subroutine is called as zero f , a , b. A more useful zero finder would not require the user to input this extra information. This more general zero finder is especially appropriate for calculators, where it is natural to simply key in a function, and awkward to then have to specify the domain.
However, it is easy to see why most zero finders require a domain. The zero finder does its work by probing the function f at various values. Then when zero f probes outside the domain of f , the code for f will return NaN, and the zero finder can continue. That is, zero f is not "punished" for making an incorrect guess. With this example in mind, it is easy to see what the result of combining a NaN with an ordinary floating-point number should be.
Similarly if one operand of a division operation is a NaN, the quotient should be a NaN. In general, whenever a NaN participates in a floating-point operation, the result is another NaN. Another approach to writing a zero solver that doesn't require the user to input a domain is to use signals. The zero-finder could install a signal handler for floating-point exceptions.
Then if f was evaluated outside its domain and raised an exception, control would be returned to the zero solver. The problem with this approach is that every language has a different method of handling signals if it has a method at all , and so it has no hope of portability.
Implementations are free to put system-dependent information into the significand. Thus there is not a unique NaN, but rather a whole family of NaNs. When a NaN and an ordinary floating-point number are combined, the result should be the same as the NaN operand. Thus if the result of a long computation is a NaN, the system-dependent information in the significand will be the information that was generated when the first NaN in the computation was generated. Actually, there is a caveat to the last statement.
If both operands are NaNs, then the result will be one of those NaNs, but it might not be the NaN that was generated first. This is much safer than simply returning the largest representable number.
So the final result is , which is safer than returning an ordinary floating-point number that is nowhere near the correct answer. The division of 0 by 0 results in a NaN. You can distinguish between getting because of overflow and getting because of division by zero by checking the status flags which will be discussed in detail in section Flags. The overflow flag will be set in the first case, the division by zero flag in the second.
The rule for determining the result of an operation that has infinity as an operand is simple: replace infinity with a finite number x and take the limit as x. When a subexpression evaluates to a NaN, the value of the entire expression is also a NaN. Here is a practical example that makes use of the rules for infinity arithmetic. Zero is represented by the exponent e min - 1 and a zero significand. Although it would be possible always to ignore the sign of zero, the IEEE standard does not do so.
When a multiplication or division involves a signed zero, the usual sign rules apply in computing the sign of the answer. Another example of the use of signed zero concerns underflow and functions that have a discontinuity at 0, such as log. Suppose that x represents a small negative number that has underflowed to zero.
Thanks to signed zero, x will be negative, so log can return a NaN. However, if there were no signed zero, the log function could not distinguish an underflowed negative number from 0, and would therefore have to return -.
Another example of a function with a discontinuity at zero is the signum function, which returns the sign of a number. Probably the most interesting use of signed zero occurs in complex arithmetic. To take a simple example, consider the equation. This is certainly true when z 0. The problem can be traced to the fact that square root is multi-valued, and there is no way to select the values so that it is continuous in the entire complex plane.
However, square root is continuous if a branch cut consisting of all negative real numbers is excluded from consideration. Signed zero provides a perfect way to resolve this problem. In fact, the natural formulas for computing will give these results.
Back to. Thus IEEE arithmetic preserves this identity for all z. Some more sophisticated examples are given by Kahan []. However, the IEEE committee decided that the advantages of utilizing the sign of zero outweighed the disadvantages.
How important is it to preserve the property. Tracking down bugs like this is frustrating and time consuming. On a more philosophical level, computer science textbooks often point out that even though it is currently impractical to prove large programs correct, designing programs with the idea of proving them often results in better code.
For example, introducing invariants is quite useful, even if they aren't going to be used as part of a proof. Floating-point code is just like any other code: it helps to have provable facts on which to depend. Similarly, knowing that 10 is true makes writing reliable floating-point code easier. If it is only true for most numbers, it cannot be used to prove anything.
The IEEE standard uses denormalized 18 numbers, which guarantee 10 , as well as other useful relations. They are the most controversial part of the standard and probably accounted for the long delay in getting approved. Most high performance hardware that claims to be IEEE compatible does not support denormalized numbers directly, but rather traps when consuming or producing denormals, and leaves it to software to simulate the IEEE standard.
The exponent e min is used to represent denormals. More formally, if the bits in the significand field are b 1 , b 2 , With denormals, x - y does not flush to zero but is instead represented by the denormalized number.
This behavior is called gradual underflow. It is easy to verify that 10 always holds when using gradual underflow. The top number line in the figure shows normalized floating-point numbers. Notice the gap between 0 and the smallest normalized number. If the result of a floating-point calculation falls into this gulf, it is flushed to zero. The bottom number line shows what happens when denormals are added to the set of floating-point numbers. The "gulf" is filled in, and when the result of a calculation is less than , it is represented by the nearest denormal.
When denormalized numbers are added to the number line, the spacing between adjacent floating-point numbers varies in a regular way: adjacent spacings are either the same length or differ by a factor of. Without denormals, the spacing abruptly changes from to , which is a factor of , rather than the orderly change by a factor of.
Because of this, many algorithms that can have large relative error for normalized numbers close to the underflow threshold are well-behaved in this range when gradual underflow is used. Large relative errors can happen even without cancellation, as the following example shows [Demmel ].
The obvious formula. A better method of computing the quotients is to use Smith's formula:. It yields 0. It is typical for denormalized numbers to guarantee error bounds for arguments all the way down to 1. When an exceptional condition like division by zero or overflow occurs in IEEE arithmetic, the default is to deliver a result and continue. The preceding sections gave examples where proceeding from an exception with these default values was the reasonable thing to do.
When any exception occurs, a status flag is also set. Implementations of the IEEE standard are required to provide users with a way to read and write the status flags. The flags are "sticky" in that once set, they remain set until explicitly cleared. Sometimes continuing execution in the face of exception conditions is not appropriate. The IEEE standard strongly recommends that implementations allow trap handlers to be installed.
Then when an exception occurs, the trap handler is called instead of setting the flag. The value returned by the trap handler will be used as the result of the operation. It is the responsibility of the trap handler to either clear or set the status flag; otherwise, the value of the flag is allowed to be undefined. The IEEE standard divides exceptions into 5 classes: overflow, underflow, division by zero, invalid operation and inexact.
There is a separate status flag for each class of exception. The meaning of the first three exceptions is self-evident. The default result of an operation that causes an invalid exception is to return a NaN, but the converse is not true. The inexact exception is raised when the result of a floating-point operation is not exact. Binary to Decimal Conversion discusses an algorithm that uses the inexact exception. There is an implementation issue connected with the fact that the inexact exception is raised so often.
If floating-point hardware does not have flags of its own, but instead interrupts the operating system to signal a floating-point exception, the cost of inexact exceptions could be prohibitive.
This cost can be avoided by having the status flags maintained by software. The first time an exception is raised, set the software flag for the appropriate class, and tell the floating-point hardware to mask off that class of exceptions. Then all further exceptions will run without interrupting the operating system. When a user resets that status flag, the hardware mask is re-enabled.
One obvious use for trap handlers is for backward compatibility. Old codes that expect to be aborted when exceptions occur can install a trap handler that aborts the process. There is a more interesting use for trap handlers that comes up when computing products such as that could potentially overflow. One solution is to use logarithms, and compute exp instead. The problem with this approach is that it is less accurate, and that it costs more than the simple expression , even if there is no overflow.
The idea is as follows. There is a global counter initialized to zero. Whenever the partial product overflows for some k , the trap handler increments the counter by one and returns the overflowed quantity with the exponent wrapped around. Similarly, if p k underflows, the counter would be decremented, and negative exponent would get wrapped around into a positive one.
When all the multiplications are done, if the counter is zero then the final product is p n. If the counter is positive, the product overflowed, if the counter is negative, it underflowed. If none of the partial products are out of range, the trap handler is never called and the computation incurs no extra cost. IEEE specifies that when an overflow or underflow trap handler is called, it is passed the wrapped-around result as an argument.
The definition of wrapped-around for overflow is that the result is computed as if to infinite precision, then divided by 2 , and then rounded to the relevant precision. For underflow, the result is multiplied by 2. The exponent is for single precision and for double precision. This is why 1. In the IEEE standard, rounding occurs whenever an operation has a result that is not exact, since with the exception of binary decimal conversion each operation is computed exactly and then rounded.
By default, rounding means round toward nearest. One application of rounding modes occurs in interval arithmetic another is mentioned in Binary to Decimal Conversion. The exact result of the addition is contained within the interval. Without rounding modes, interval arithmetic is usually implemented by computing and , where is machine epsilon. Since the result of an operation in interval arithmetic is an interval, in general the input to an operation will also be an interval.
When a floating-point calculation is performed using interval arithmetic, the final answer is an interval that contains the exact result of the calculation. This is not very helpful if the interval turns out to be large as it often does , since the correct answer could be anywhere in that interval. Interval arithmetic makes more sense when used in conjunction with a multiple precision floating-point package. The calculation is first performed with some precision p. If interval arithmetic suggests that the final answer may be inaccurate, the computation is redone with higher and higher precisions until the final interval is a reasonable size.
The IEEE standard has a number of flags and modes. As discussed above, there is one status flag for each of the five exceptions: underflow, overflow, division by zero, invalid operation and inexact.
It is strongly recommended that there be an enable mode bit for each of the five exceptions. This section gives some simple examples of how these modes and flags can be put to good use. A more sophisticated example is discussed in the section Binary to Decimal Conversion. Consider writing a subroutine to compute x n , where n is an integer. In the second expression these are exact i. Unfortunately, these is a slight snag in this strategy.
If PositivePower x, -n underflows, then either the underflow trap handler will be called, or else the underflow status flag will be set. This is incorrect, because if x - n underflows, then x n will either overflow or be in range. It simply turns off the overflow and underflow trap enable bits and saves the overflow and underflow status bits. If neither the overflow nor underflow status bit is set, it restores them together with the trap enable bits.
Another example of the use of flags occurs when computing arccos via the formula. The solution to this problem is straightforward. Simply save the value of the divide by zero flag before computing arccos, and then restore its old value after the computation. The design of almost every aspect of a computer system requires knowledge about floating-point. Computer architectures usually have floating-point instructions, compilers must generate those floating-point instructions, and the operating system must decide what to do when exception conditions are raised for those floating-point instructions.
Computer system designers rarely get guidance from numerical analysis texts, which are typically aimed at users and writers of software, not at computer designers. As an example of how plausible design decisions can lead to unexpected behavior, consider the following BASIC program. This example will be analyzed in the next section. Incidentally, some people think that the solution to such anomalies is never to compare floating-point numbers for equality, but instead to consider them equal if they are within some error bound E.
This is hardly a cure-all because it raises as many questions as it answers. What should the value of E be?
It is quite common for an algorithm to require a short burst of higher precision in order to produce accurate results. As discussed in the section Proof of Theorem 4 , when b 2 4 ac , rounding error can contaminate up to half the digits in the roots computed with the quadratic formula. By performing the subcalculation of b 2 - 4 ac in double precision, half the double precision bits of the root are lost, which means that all the single precision bits are preserved. The computation of b 2 - 4 ac in double precision when each of the quantities a , b , and c are in single precision is easy if there is a multiplication instruction that takes two single precision numbers and produces a double precision result.
In order to produce the exactly rounded product of two p -digit numbers, a multiplier needs to generate the entire 2 p bits of product, although it may throw bits away as it proceeds.
Thus, hardware to compute a double precision product from single precision operands will normally be only a little more expensive than a single precision multiplier, and much cheaper than a double precision multiplier. Despite this, modern instruction sets tend to provide only instructions that produce a result of the same precision as the operands.
If an instruction that combines two single precision operands to produce a double precision product was only useful for the quadratic formula, it wouldn't be worth adding to an instruction set.
However, this instruction has many other uses. Consider the problem of solving a system of linear equations,. Suppose that a solution x 1 is computed by some method, perhaps Gaussian elimination. There is a simple way to improve the accuracy of the result called iterative improvement. First compute. Note that if x 1 is an exact solution, then is the zero vector, as is y.
Then y x 1 - x , so an improved estimate for the solution is. The three steps 12 , 13 , and 14 can be repeated, replacing x 1 with x 2 , and x 2 with x 3. For more information, see [Golub and Van Loan ]. When performing iterative improvement, is a vector whose elements are the difference of nearby inexact floating-point numbers, and so can suffer from catastrophic cancellation.
Once again, this is a case of computing the product of two single precision numbers A and x 1 , where the full double precision result is needed. To summarize, instructions that multiply two floating-point numbers and return a product with twice the precision of the operands make a useful addition to a floating-point instruction set. Some of the implications of this for compilers are discussed in the next section.
The interaction of compilers and floating-point is discussed in Farnum [], and much of the discussion in this section is taken from that paper. Ideally, a language definition should define the semantics of the language precisely enough to prove statements about programs. While this is usually true for the integer part of a language, language definitions often have a large grey area when it comes to floating-point.
Perhaps this is due to the fact that many language designers believe that nothing can be proven about floating-point, since it entails rounding error. If so, the previous sections have demonstrated the fallacy in this reasoning. This section discusses some common grey areas in language definitions, including suggestions about how to deal with them.
Remarkably enough, some languages don't clearly specify that if x is a floating-point variable with say a value of 3. For example Ada, which is based on Brown's model, seems to imply that floating-point arithmetic only has to satisfy Brown's axioms, and thus expressions can have one of many possible values.
Thinking about floating-point in this fuzzy way stands in sharp contrast to the IEEE model, where the result of each floating-point operation is precisely defined. In the IEEE model, we can prove that 3. In Brown's model, we cannot. Another ambiguity in most language definitions concerns what happens on overflow, underflow and other exceptions.
The IEEE standard precisely specifies the behavior of exceptions, and so languages that use the standard as a model can avoid any ambiguity on this point. Another grey area concerns the interpretation of parentheses. Due to roundoff errors, the associative laws of algebra do not necessarily hold for floating-point numbers. The importance of preserving parentheses cannot be overemphasized. The algorithms presented in theorems 3, 4 and 6 all depend on it. A language definition that does not require parentheses to be honored is useless for floating-point calculations.
Subexpression evaluation is imprecisely defined in many languages. Suppose that ds is double precision, but x and y are single precision. There are two ways to deal with this problem, neither of which is completely satisfactory. The first is to require that all variables in an expression have the same type.
This is the simplest solution, but has some drawbacks. First of all, languages like Pascal that have subrange types allow mixing subrange variables with integer variables, so it is somewhat bizarre to prohibit mixing single and double precision variables.
Another problem concerns constants. In the expression 0. Now suppose the programmer decides to change the declaration of all the floating-point variables from single to double precision.
The programmer will have to hunt down and change every floating-point constant. The second approach is to allow mixed expressions, in which case rules for subexpression evaluation must be provided. There are a number of guiding examples. The original definition of C required that every floating-point expression be computed in double precision [Kernighan and Ritchie ].
This leads to anomalies like the example at the beginning of this section. The expression 3. This suggests that computing every expression in the highest precision available is not a good rule. Another guiding example is inner products. If the inner product has thousands of terms, the rounding error in the sum can become substantial.
One way to reduce this rounding error is to accumulate the sums in double precision this will be discussed in more detail in the section Optimizers. If the multiplication is done in single precision, than much of the advantage of double precision accumulation is lost, because the product is truncated to single precision just before being added to a double precision variable.
A rule that covers both of the previous two examples is to compute an expression in the highest precision of any variable that occurs in that expression.
However, this rule is too simplistic to cover all cases cleanly. A more sophisticated subexpression evaluation rule is as follows. First assign each operation a tentative precision, which is the maximum of the precisions of its operands. This assignment has to be carried out from the leaves to the root of the expression tree.
What is UV radiation? What is at stake? Two types of UV light are proven to contribute to the risk for skin cancer: Ultraviolet A UVA has a longer wavelength, and is associated with skin aging. Ultraviolet B UVB has a shorter wavelength and is associated with skin burning. What you need to know. A majority of nonmelanoma skin cancers NMSC and a large percentage of melanomas are associated with exposure to UV radiation from the sun. UV exposure is a powerful attack on the skin, creating damage that can range from premature wrinkles to dangerous skin cancer.
Damage from UV exposure is cumulative and increases your skin cancer risk over time. The unrepaired damage builds up over time and triggers mutations that cause skin cells to multiply rapidly.
That can lead to malignant tumors. The degree of damage depends on the intensity of UV rays and the length of time your skin has been exposed without protection. Location is also a factor. If you live where the sun is strong year-round, your exposure level and risk increases.
You can easily reduce your likelihood of developing skin cancer by taking care to protect yourself against UV radiation. The sister act claimed their first set of the match, allowing Laguna Beach to settle into the contest with a lead after the first rotation of play.
The pairing of Sydnee Spirlin and Sara Miller won once. Support our sports coverage by becoming a digital subscriber. For more sports stories, visit latimes.