|
public static LeftShift ( |
||
| value | The value whose bits are to be shifted. | |
| shift | int | The number of bits to shift |
| return | ||
/// <summary>
/// Scales the given double-precision number by multiplying and then shifting it.
/// </summary>
/// <param name="value"> The value to scale. </param>
/// <param name="multiplier"> The multiplier. </param>
/// <param name="shift"> The power of two scale factor. </param>
/// <returns> A BigInteger containing the result of multiplying <paramref name="value"/> by
/// <paramref name="multiplier"/> and then shifting left by <paramref name="shift"/> bits. </returns>
private static BigInteger ScaleToInteger(double value, BigInteger multiplier, int shift)
{
long bits = BitConverter.DoubleToInt64Bits(value);
// Extract the base-2 exponent.
var base2Exponent = (int)((bits & 0x7FF0000000000000) >> 52) - 1023;
// Extract the mantissa.
long mantissa = bits & 0xFFFFFFFFFFFFF;
if (base2Exponent > -1023)
{
mantissa |= 0x10000000000000;
base2Exponent -= 52;
}
else
{
// Denormals.
base2Exponent -= 51;
}
// Extract the sign bit.
if (bits < 0)
{
mantissa = -mantissa;
}
var result = new BigInteger(mantissa);
result = BigInteger.Multiply(result, multiplier);
shift += base2Exponent;
result = BigInteger.LeftShift(result, shift);
return(result);
}
/// <summary>
/// Returns a new instance BigInteger structure from a 64-bit double precision floating
/// point value.
/// </summary>
/// <param name="value"> A 64-bit double precision floating point value. </param>
/// <returns> The corresponding BigInteger value. </returns>
public static BigInteger FromDouble(double value)
{
long bits = XBitConverter.DoubleToInt64Bits(value);
// Extract the base-2 exponent.
var base2Exponent = (int)((bits & 0x7FF0000000000000) >> 52) - 1023;
// Extract the mantissa.
long mantissa = bits & 0xFFFFFFFFFFFFF;
if (base2Exponent > -1023)
{
mantissa |= 0x10000000000000;
base2Exponent -= 52;
}
else
{
// Denormals.
base2Exponent -= 51;
}
// Extract the sign bit.
if (bits < 0)
{
mantissa = -mantissa;
}
return(BigInteger.LeftShift(new BigInteger(mantissa), base2Exponent));
}
/// <summary>
/// Modifies the given values so they are suitable for passing to Quorem.
/// </summary>
/// <param name="dividend"> The number that will be divided. </param>
/// <param name="divisor"> The number to divide by. </param>
public static void SetupQuorum(ref BigInteger dividend, ref BigInteger divisor)
{
var leadingZeroCount = CountLeadingZeroBits(divisor.bits[divisor.wordCount - 1]);
if (leadingZeroCount < 4 || leadingZeroCount > 28)
{
dividend = BigInteger.LeftShift(dividend, 8);
divisor = BigInteger.LeftShift(divisor, 8);
}
}
/// <summary>
/// Equivalent to BigInteger.Pow but with integer arguments.
/// </summary>
/// <param name="radix"> The number to be raised to a power. </param>
/// <param name="exponent"> The number that specifies the power. </param>
/// <returns> The number <paramref name="radix"/> raised to the power
/// <paramref name="exponent"/>. </returns>
public static BigInteger Pow(int radix, int exponent)
{
if (radix < 0 || radix > 36)
{
throw new ArgumentOutOfRangeException("radix");
}
if (exponent < 0)
{
throw new ArgumentOutOfRangeException("exponent");
}
if (radix == 10 && exponent < integerPowersOfTen.Length)
{
// Use a table for quick lookup of powers of 10.
return(new BigInteger(integerPowersOfTen[exponent]));
}
else if (radix == 2)
{
// Power of two is easy.
return(BigInteger.LeftShift(BigInteger.One, exponent));
}
// Special cases.
switch (exponent)
{
case 0:
return(BigInteger.One);
case 1:
return(new BigInteger(radix));
case 2:
return(new BigInteger(radix * radix));
case 3:
return(new BigInteger(radix * radix * radix));
}
// Use recursion to calculate the result.
if ((exponent & 1) == 1)
{
// Exponent is odd.
var temp = Pow(radix, exponent / 2);
return(BigInteger.MultiplyAdd(BigInteger.Multiply(temp, temp), radix, 0));
}
else
{
// Exponent is even.
var temp = Pow(radix, exponent / 2);
return(BigInteger.Multiply(temp, temp));
}
}
/// <summary>
/// Converts a number to a string.
/// </summary>
/// <param name="value"> The value to convert to a string. </param>
/// <param name="radix"> The base of the number system to convert to. </param>
/// <param name="numberFormatInfo"> The number format style to use. </param>
/// <param name="style"> The type of formatting to apply. </param>
/// <param name="precision">
/// This value is dependent on the formatting style:
/// Regular - this value has no meaning.
/// Precision - the number of significant figures to display.
/// Fixed - the number of figures to display after the decimal point.
/// Exponential - the number of figures to display after the decimal point.
/// </param>
internal static string ToString(double value, int radix, System.Globalization.NumberFormatInfo numberFormatInfo, Style style, int precision = 0)
{
// Handle NaN.
if (double.IsNaN(value))
{
return(numberFormatInfo.NaNSymbol); // "NaN"
}
// Handle zero.
if (value == 0.0)
{
switch (style)
{
case Style.Regular:
return("0");
case Style.Precision:
return("0" + numberFormatInfo.NumberDecimalSeparator + new string('0', precision - 1));
case Style.Fixed:
if (precision == 0)
{
return("0");
}
return("0" + numberFormatInfo.NumberDecimalSeparator + new string('0', precision));
case Style.Exponential:
if (precision <= 0)
{
return("0" + exponentSymbol + numberFormatInfo.PositiveSign + "0");
}
return("0" + numberFormatInfo.NumberDecimalSeparator + new string('0', precision) + exponentSymbol + numberFormatInfo.PositiveSign + "0");
}
}
var result = new System.Text.StringBuilder(18);
// Handle negative numbers.
if (value < 0.0)
{
value = -value;
result.Append(numberFormatInfo.NegativeSign);
}
// Handle infinity.
if (double.IsPositiveInfinity(value))
{
result.Append(numberFormatInfo.PositiveInfinitySymbol); // "Infinity"
return(result.ToString());
}
// Extract the base-2 exponent.
var bits = new DoubleBits()
{
DoubleValue = value
};
int base2Exponent = (int)(bits.LongValue >> MantissaExplicitBits);
// Extract the mantissa.
long mantissa = bits.LongValue & MantissaMask;
// Correct the base-2 exponent.
if (base2Exponent == 0)
{
// This is a denormalized number.
base2Exponent = base2Exponent - ExponentBias - MantissaExplicitBits + 1;
}
else
{
// This is a normal number.
base2Exponent = base2Exponent - ExponentBias - MantissaExplicitBits;
// Add the implicit bit.
mantissa |= MantissaImplicitBit;
}
// Remove any trailing zeros.
int trailingZeroBits = CountTrailingZeroBits((ulong)mantissa);
mantissa >>= trailingZeroBits;
base2Exponent += trailingZeroBits;
// Calculate the logarithm of the number.
int exponent;
if (radix == 10)
{
exponent = (int)Math.Floor(Math.Log10(value));
// We need to calculate k = floor(log10(x)).
// log(x) ~=~ log(1.5) + (x-1.5)/1.5 (taylor series approximation)
// log10(x) ~=~ log(1.5) / log(10) + (x - 1.5) / (1.5 * log(10))
// d = x * 2^l (1 <= x < 2)
// log10(d) = l * log10(2) + log10(x)
// log10(d) ~=~ l * log10(2) + (x - 1.5) * (1 / (1.5 * log(10))) + log(1.5) / log(10)
// log10(d) ~=~ l * 0.301029995663981 + (x - 1.5) * 0.289529654602168 + 0.1760912590558
// The last term (0.1760912590558) is rounded so that k = floor(log10(x)) or
// k = floor(log10(x)) + 1 (i.e. it's the exact value or one higher).
//double log10;
//if ((int)(bits.LongValue >> MantissaExplicitBits) == 0)
//{
// // The number is denormalized.
// int mantissaShift = CountLeadingZeroBits((ulong)mantissa) - (64 - MantissaImplicitBits);
// bits.LongValue = (mantissa << mantissaShift) & MantissaMask |
// ((long)ExponentBias << MantissaExplicitBits);
// // Calculate an overestimate of log-10 of the value.
// log10 = (bits.DoubleValue - 1.5) * 0.289529654602168 + 0.1760912590558 +
// (base2Exponent - mantissaShift) * 0.301029995663981;
//}
//else
//{
// // Set the base-2 exponent to biased zero.
// bits.LongValue = (bits.LongValue & ~ExponentMask) | ((long)ExponentBias << MantissaExplicitBits);
// // Calculate an overestimate of log-10 of the value.
// log10 = (bits.DoubleValue - 1.5) * 0.289529654602168 + 0.1760912590558 + base2Exponent * 0.301029995663981;
//}
//// (int)Math.Floor(log10)
//exponent = (int)log10;
//if (log10 < 0 && log10 != exponent)
// exponent--;
//if (exponent >= 0 && exponent < tens.Length)
//{
// if (value < tens[exponent])
// exponent--;
//}
}
else
{
exponent = (int)Math.Floor(Math.Log(value, radix));
}
if (radix == 10 && style == Style.Regular)
{
// Do we have a small integer?
if (base2Exponent >= 0 && exponent <= 14)
{
// Yes.
for (int i = exponent; i >= 0; i--)
{
double scaleFactor = tens[i];
int digit = (int)(value / scaleFactor);
result.Append((char)(digit + '0'));
value -= digit * scaleFactor;
}
return(result.ToString());
}
}
// toFixed acts like toString() if the exponent is >= 21.
if (style == Style.Fixed && exponent >= 21)
{
style = Style.Regular;
}
// Calculate the exponent thresholds.
int lowExponentThreshold = int.MinValue;
if (radix == 10 && style != Style.Fixed)
{
lowExponentThreshold = -7;
}
if (style == Style.Exponential)
{
lowExponentThreshold = -1;
}
int highExponentThreshold = int.MaxValue;
if (radix == 10 && style == Style.Regular)
{
highExponentThreshold = 21;
}
if (style == Style.Precision)
{
highExponentThreshold = precision;
}
if (style == Style.Exponential)
{
highExponentThreshold = 0;
}
// Calculate the number of bits per digit.
double bitsPerDigit = radix == 10 ? 3.322 : Math.Log(radix, 2);
// Calculate the maximum number of digits to output.
// We add 7 so that there is enough precision to distinguish halfway numbers.
int maxDigitsToOutput = radix == 10 ? 22 : (int)Math.Floor(53 / bitsPerDigit) + 7;
// Calculate the number of integral digits, or if negative, the number of zeros after
// the decimal point.
int integralDigits = exponent + 1;
// toFixed with a low precision causes rounding.
if (style == Style.Fixed && precision <= -integralDigits)
{
int diff = (-integralDigits) - (precision - 1);
maxDigitsToOutput += diff;
exponent += diff;
integralDigits += diff;
}
// Output any leading zeros.
bool decimalPointOutput = false;
if (integralDigits <= 0 && integralDigits > lowExponentThreshold + 1)
{
result.Append('0');
if (integralDigits < 0)
{
result.Append(numberFormatInfo.NumberDecimalSeparator);
decimalPointOutput = true;
result.Append('0', -integralDigits);
}
}
// We need to calculate the integers "scaledValue" and "divisor" such that:
// value = scaledValue / divisor * 10 ^ exponent
// 1 <= scaledValue / divisor < 10
BigInteger scaledValue = new BigInteger(mantissa);
BigInteger divisor = BigInteger.One;
BigInteger multiplier = BigInteger.One;
if (exponent > 0)
{
// Number is >= 10.
divisor = BigInteger.Multiply(divisor, BigInteger.Pow(radix, exponent));
}
else if (exponent < 0)
{
// Number is < 1.
multiplier = BigInteger.Pow(radix, -exponent);
scaledValue = BigInteger.Multiply(scaledValue, multiplier);
}
// Scale the divisor so it is 74 bits ((21 digits + 1 digit for rounding) * approx 3.322 bits per digit).
int powerOfTwoScaleFactor = (radix == 10 ? 74 : (int)Math.Ceiling(maxDigitsToOutput * bitsPerDigit)) - divisor.BitCount;
divisor = BigInteger.LeftShift(divisor, powerOfTwoScaleFactor);
scaledValue = BigInteger.LeftShift(scaledValue, powerOfTwoScaleFactor + base2Exponent);
// Calculate the error.
BigInteger errorDelta = BigInteger.Zero;
int errorPowerOfTen = int.MinValue;
switch (style)
{
case Style.Regular:
errorDelta = ScaleToInteger(CalculateError(value), multiplier, powerOfTwoScaleFactor - 1);
break;
case Style.Precision:
errorPowerOfTen = integralDigits - precision;
break;
case Style.Fixed:
errorPowerOfTen = -precision;
break;
case Style.Exponential:
if (precision < 0)
{
errorDelta = ScaleToInteger(CalculateError(value), multiplier, powerOfTwoScaleFactor - 1);
}
else
{
errorPowerOfTen = integralDigits - precision - 1;
}
break;
default:
throw new ArgumentException(nameof(style));
}
if (errorPowerOfTen != int.MinValue)
{
errorDelta = multiplier;
if (errorPowerOfTen > 0)
{
errorDelta = BigInteger.Multiply(errorDelta, BigInteger.Pow(radix, errorPowerOfTen));
}
errorDelta = BigInteger.LeftShift(errorDelta, powerOfTwoScaleFactor - 1);
if (errorPowerOfTen < 0)
{
// We would normally divide by the power of 10 here, but division is extremely
// slow so we multiply everything else instead.
//errorDelta = BigInteger.Divide(errorDelta, BigInteger.Pow(radix, -errorPowerOfTen));
var errorPowerOfTenMultiplier = BigInteger.Pow(radix, -errorPowerOfTen);
scaledValue = BigInteger.Multiply(scaledValue, errorPowerOfTenMultiplier);
divisor = BigInteger.Multiply(divisor, errorPowerOfTenMultiplier);
BigInteger.SetupQuorum(ref scaledValue, ref divisor, ref errorDelta);
}
}
// Shrink the error in the case where ties are resolved towards the value with the
// least significant bit set to zero.
if ((BitConverter.DoubleToInt64Bits(value) & 1) == 1)
{
errorDelta.InPlaceDecrement();
}
// Cache half the divisor.
BigInteger halfDivisor = BigInteger.RightShift(divisor, 1);
// Output the digits.
int zeroCount = 0;
int digitsOutput = 0;
bool rounded = false, scientificNotation = false;
for (; digitsOutput < maxDigitsToOutput && rounded == false; digitsOutput++)
{
// Calculate the next digit.
var digit = (int)BigInteger.Quorem(ref scaledValue, divisor);
if (BigInteger.Compare(scaledValue, errorDelta) <= 0 && BigInteger.Compare(scaledValue, halfDivisor) < 0)
{
// Round down.
rounded = true;
}
else if (BigInteger.Compare(BigInteger.Subtract(divisor, scaledValue), errorDelta) <= 0)
{
// Round up.
rounded = true;
digit++;
if (digit == radix)
{
digit = 1;
exponent++;
integralDigits++;
}
}
if (digit > 0 || decimalPointOutput == false)
{
// Check if the decimal point should be output.
if (decimalPointOutput == false && (scientificNotation == true || digitsOutput == integralDigits))
{
result.Append(numberFormatInfo.NumberDecimalSeparator);
decimalPointOutput = true;
}
// Output any pent-up zeros.
if (zeroCount > 0)
{
result.Append('0', zeroCount);
zeroCount = 0;
}
// Output the next digit.
if (digit < 10)
{
result.Append((char)(digit + '0'));
}
else
{
result.Append((char)(digit - 10 + 'a'));
}
}
else
{
zeroCount++;
}
// Check whether the number should be displayed in scientific notation (we cannot
// determine this up front for large exponents because the number might get rounded
// up to cross the threshold).
if (digitsOutput == 0 && (exponent <= lowExponentThreshold || exponent >= highExponentThreshold))
{
scientificNotation = true;
}
scaledValue = BigInteger.MultiplyAdd(scaledValue, radix, 0);
errorDelta = BigInteger.MultiplyAdd(errorDelta, radix, 0);
}
// Add any extra zeros on the end, if necessary.
if (scientificNotation == false && integralDigits > digitsOutput)
{
result.Append('0', integralDigits - digitsOutput);
digitsOutput = integralDigits;
}
// Most of the styles output redundent zeros.
int redundentZeroCount = 0;
switch (style)
{
case Style.Precision:
redundentZeroCount = zeroCount + precision - digitsOutput;
break;
case Style.Fixed:
redundentZeroCount = precision - (digitsOutput - zeroCount - integralDigits);
break;
case Style.Exponential:
redundentZeroCount = precision - (digitsOutput - zeroCount) + 1;
break;
}
if (redundentZeroCount > 0)
{
if (decimalPointOutput == false)
{
result.Append(numberFormatInfo.NumberDecimalSeparator);
}
result.Append('0', redundentZeroCount);
}
if (scientificNotation == true)
{
// Add the exponent on the end.
result.Append(exponentSymbol);
if (exponent > 0)
{
result.Append(numberFormatInfo.PositiveSign);
}
result.Append(exponent);
}
return(result.ToString());
}
/// <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);
}