The power handling and much of the bit manipulations are motivated by the discussion in Section 2. Note that in production code, you must handle the case when the denominator of input r is zero.

## Eberly, David H GPGPU Programming for Games and Science

The example here does not do so. Modify the pseudocode to handle such a rational input. In Equation 2. Normal 8-bit numbers are converted to normal bit numbers in a trivial manner. The biased exponent for the 8-bit number is adjusted to become a biased exponent for the bit number. The trailing bits for the 8-bit number are copied to the correct location in the trailing bits for the bit number.

Subnormal 8-bit numbers are also converted to normal bit numbers. The conversion is the following:. By inspecting and testing several cases, verify that the conversions are correct. The pattern is general for conversion to a wider format. Round-to-nearest is used with ties-to-even.

A wide-format NaN is mapped to a narrow-format NaN, but if the wide-format payload has more 1-valued bits than can be stored in the narrow-format payload, there will be a loss of information. The mapping from wider to nar- rower format is illustrated with grayscale bars and text indicating how to round. All labeled values are exactly representable in the wide format, but some are not exactly representable in the narrow format. For example, nar-avr-min-normal-zero is the average of nar-zero and nar-min-subnormal.

Even though the two inputs are exactly representable in the narrow format, the average is not. This is not a problem, because the comparisons made during conversion are all in the wide-format number system. Using encodings in the wide-format number system, let xzero be the posi- tive zero for the narrow format and let xsub0 be the minimum subnormal for the narrow format. All such conversions are inexact except for zero itself. Again, all such conversion are inexact except for xsub0 itself.

## GPGPU Programming for Games and Science

Similarly, let xnor1 be the maximum normal for the narrow format. All such conversions are inexact except for xnor1 itself. The conversions are all inexact. The wide-format numbers in the open interval xsub0 , xnor1 are converted to narrow-format numbers using round-to-nearest with ties-to-even. Many of the conversions are inexact, but some are exact. The open interval in terms of wide-format numbers written as subnormals is the closed interval [0.

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The carry-out is to the low-order bit of the biased exponent. Regarding an implementation, it is convenient to generate i and f in a canonical format. The integer 1t represents the t with a prepended 1, the result containing eleven bits. The comparison of 0. The input is 1. Both input and output are in normal form, so there is no need to prepend one to t.

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The same issue arises about a carry-out to the exponent when rounding. Modify the source code for conversion from bit encodings to 8-bit encodings by inserting state- ments that set the appropriate bits of the status words when exceptions occur. In particular, the inexact-operation bit must be set when a bit number can- not be exactly represented as an 8-bit number. The arbitrary choice was made to set the 8-payload to 1. You cannot rely on a 1-valued low-order pay- load bit to indicate the inexact payload conversion, because a bit payload of will map to an 8-bit payload of , and the 1-bit in the bit payload does not correspond to an inexact payload conversion from bit to bit.

Modify the source code to support this choice. The pattern is general for conversion to a narrower format.

The corresponding narrow subnormal number is 0. If the input is 1. The reason for the right shift equation is motivated by the example provided previously. The minimum for the subnormal interval is. Think of of 1t embedded in a large set of bits so that shifting does not lose any bits.

The right shift that moves the fractional part to the high-order bit of the k1 -bit block must be k1. The total left shifting is therefore. Consider the case when the input and output are both normal numbers.

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The wide input is 1. The pseudocode is shown in Listing 2. The other bits are lost. Hint: Think about this in terms of code coverage. Division can be implemented as described previously, using a straightforward division of binary numbers similar to what you do for long division of integers. Other division approaches are possible. Using a minimax algorithm see Section 3. A more interesting implementation of division in hardware uses multiplica- tive division [15].

With carefully chosen factors, we can itera- tively drive the denominator yf0. For example, Listing 2. The exact value is irrational but may be expanded to as many binary places as is shown. Generally, deriving mathematical approximations to functions with correct rounding is a technical challenge. FPU hardware can provide fast computation by using registers with higher precision than that of the inputs to the func- tions.

## Eberly, GPGPU Programming for Games and Science, 1e

However, you may consider trading accuracy for speed. The standard mathematics library that ships with Microsoft Visual Studio has several functions supported by an FPU, shown in Listing 2. The new formula is mathematically equivalent to the old formula, but now the denominator has a sum of nearly equal values, which avoids the cancellation.

The basic illustration of subtractive cancellation is shown in Listing 2. Observe that the polyno- mial value at that root is nearly zero. This is clear by observing that the polynomial values at the neighbors have opposite signs and the magnitude of the polynomial value at the estimated root is smaller than the magnitudes of the polynomial values at the neighbors.

Here are sev- eral questions for investigation. Is it ever possible for the Newton iterates to cycle, thus preventing convergence to an estimated root?

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Let rd be the next-down neighbor of r and let ru be the next-up neighbor of r. The original formula estimates a root of 0. Moreover, the magnitude of the polynomial value at the estimate root is smaller than the magnitudes of the polynomial values at the neighbors.