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CodeTuned Time 1.05 1.40 4.17 12.9
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Language C++ Java Python Visual Basic
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Straight Time 4.06 2.54 8.16 24.0
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Note: The results shown assume that the cache is hit twice for each time it s set.
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The success of the cache depends on the relative costs of accessing a cached element, creating an uncached element, and saving a new element in the cache.
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de Complete
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26. Code-Tuning Techniques
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Success also depends on how often the cached information is requested. In some cases, success might also depend on caching done by the hardware. Generally, the more it costs to generate a new element and the more times the same information is requested, the more valuable a cache is. The cheaper it is to access a cached element and save new elements in the cache, the more valuable a cache is. As with other optimization techniques, caching adds complexity and tends to be error prone.
26.4 Expressions
9 CROSS-REFERENCE
0 more information on
expressions, see Section 19.1, Boolean Expressions.
Much of the work in a program is done inside mathematical or logical expressions. Complicated expressions tend to be expensive, so this section looks at ways to make them cheaper.
Exploit Algebraic Identities
You can use algebraic identities to replace costly operations with cheaper ones. For example, the following expressions are logically equivalent:
not a and not B not (a or B)
If you choose the second expression instead of the first, you can save a not operation. Although the savings from avoiding a single not operation are probably inconsequential, the general principle is powerful. Jon Bentley describes a program that tested whether sqrt(x) < sqrt(y) (1982). Since sqrt(x) is less than sqrt(y) only when x is less than y, you can replace the first test with x < y. Given the cost of the sqrt() routine, you d expect the savings to be dramatic, and they are. Here are the results:
CodeTuned Time 0.010 0.220 0.401
Language C++ Visual Basic Python
Straight Time 7.43 4.59 4.21
Time Savings 99.9% 95% 90%
Performance Ratio 750:1 20:1 10:1
Use Strength Reduction
As mentioned earlier, strength reduction means replacing an expensive operation with a cheaper one. Here are some possible substitutions:
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26. Code-Tuning Techniques
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Replace multiplication with addition. Replace exponentiation with multiplication. Replace trigonometric routines with their trigonometric identities. Replace longlong integers with longs or ints (but watch for performance issues associated with using native-length vs. non-native length integers) Replace floating-point numbers with fixed-point numbers or integers. Replace double-precision floating points with single-precision numbers. Replace integer multiplication-by-two and division-by-two with shift operations.
Here is a detailed example. Suppose you have to evaluate a polynomial. If you re rusty on polynomials, they re the things that look like Ax2 + Bx + C The letters A, B, and C are coefficients, and x is a variable. General code to evaluate an nth-order polynomial looks like this:
Visual Basic Example of Evaluating a Polynomial
value = coefficient( 0 ) For power = 1 To order value = value + coefficient( power ) * x^power Next
If you re thinking about strength reduction, you ll look at the exponentiation operator with a jaundiced eye. One solution would be to replace the exponentiation with a multiplication on each pass through the loop, which is analogous to the strength-reduction case a few sections ago in which a multiplication was replaced with an addition. Here s how the reduced-strength polynomial evaluation would look:
Visual Basic Example of a Reduced-Strength Method of Evaluating a Polynomial
value = coefficient( 0 ) powerOfX = x For power = 1 to order value = value + coefficient( power ) * powerOfX powerOfX = powerOfX * x Next
This produces a noticeable advantage if you re working with second-order polynomials (polynomials in which the highest-power term is squared) or higherorder polynomials.
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Language Python Visual Basic
Straight Time 3.24 6.26
CodeTuned Time 2.60 0.160
Time Savings 20% 97%
Performance Ratio 1:1 40:1
If you re serious about strength reduction, you still won t care for those two floating-point multiplications. The strength-reduction principle suggests that you can further reduce the strength of the operations in the loop by accumulating powers rather than multiplying them each time. Here s that code:
Visual Basic Example of Further Reducing the Strength Required to Evaluate a Polynomial
value = 0 For power = order to 1 Step -1 value = ( value + coefficient( power ) ) * x Next value = value + coefficient( 0 )
This method eliminates the extra powerOfX variable and replaces the two multiplications in each pass through the loop with one.
Savings over First Optimization 3% -94%
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