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public static Abs ( |
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| b | A number. | |
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/// <summary>
/// Modifies the initial estimate until the closest double-precision number to the desired
/// value is found.
/// </summary>
/// <param name="initialEstimate"> The initial estimate. Assumed to be very close to the
/// result. </param>
/// <param name="base10Exponent"> The power-of-ten scale factor. </param>
/// <param name="desiredValue"> The desired value, already scaled using the power-of-ten
/// scale factor. </param>
/// <returns> The closest double-precision number to the desired value. If there are two
/// such values, the one with the least significant bit set to zero is returned. </returns>
private static double RefineEstimate(double initialEstimate, int base10Exponent, BigInteger desiredValue)
{
// Numbers with 16 digits or more are tricky because rounding error can cause the
// result to be out by one or more ULPs (units in the last place).
// The algorithm is as follows:
// 1. Use the initially calculated result as an estimate.
// 2. Create a second estimate by modifying the estimate by one ULP.
// 3. Calculate the actual value of both estimates to precision X (using arbitrary
// precision arithmetic).
// 4. If they are both above the desired value then decrease the estimates by 1
// ULP and goto step 3.
// 5. If they are both below the desired value then increase up the estimates by
// 1 ULP and goto step 3.
// 6. One estimate must now be above the desired value and one below.
// 7. If one is estimate is clearly closer to the desired value than the other,
// then return that estimate.
// 8. Increase the precision by 32 bits.
// 9. If the precision is less than or equal to 160 bits goto step 3.
// 10. Assume that the estimates are equally close to the desired value; return the
// value with the least significant bit equal to 0.
int direction = double.IsPositiveInfinity(initialEstimate) ? -1 : 1;
int precision = 32;
// Calculate the candidate value by modifying the last bit.
double result = initialEstimate;
double result2 = AddUlps(result, direction);
// Figure out our multiplier. Either base10Exponent is positive, in which case we
// multiply actual1 and actual2, or it's negative, in which case we multiply
// desiredValue.
BigInteger multiplier = BigInteger.One;
if (base10Exponent < 0)
{
multiplier = BigInteger.Pow(10, -base10Exponent);
}
else if (base10Exponent > 0)
{
desiredValue = BigInteger.Multiply(desiredValue, BigInteger.Pow(10, base10Exponent));
}
while (precision <= 160)
{
// Scale the candidate values to a big integer.
var actual1 = ScaleToInteger(result, multiplier, precision);
var actual2 = ScaleToInteger(result2, multiplier, precision);
// Calculate the differences between the candidate values and the desired value.
var baseline = BigInteger.LeftShift(desiredValue, precision);
var diff1 = BigInteger.Subtract(actual1, baseline);
var diff2 = BigInteger.Subtract(actual2, baseline);
if (diff1.Sign == direction && diff2.Sign == direction)
{
// We're going the wrong way!
direction = -direction;
result2 = AddUlps(result, direction);
}
else if (diff1.Sign == -direction && diff2.Sign == -direction)
{
// Going the right way, but need to go further.
result = result2;
result2 = AddUlps(result, direction);
}
else
{
// Found two values that bracket the actual value.
// If one candidate value is closer to the actual value by at least 2 (one
// doesn't cut it because of the integer division) then use that value.
diff1 = BigInteger.Abs(diff1);
diff2 = BigInteger.Abs(diff2);
if (BigInteger.Compare(diff1, BigInteger.Subtract(diff2, BigInteger.One)) < 0)
{
return(result);
}
if (BigInteger.Compare(diff2, BigInteger.Subtract(diff1, BigInteger.One)) < 0)
{
return(result2);
}
// Not enough precision to determine the correct answer, or it's a halfway case.
// Increase the precision.
precision += 32;
}
// If result2 is NaN then we have gone too far.
if (double.IsNaN(result2) == true)
{
return(result);
}
}
// Even with heaps of precision there is no clear winner.
// Assume this is a halfway case: choose the floating-point value with its least
// significant bit equal to 0.
return((BitConverter.DoubleToInt64Bits(result) & 1) == 0 ? result : result2);
}
