/// <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 the numeric value of the current BigInteger object to its equivalent string /// representation. /// </summary> /// <returns> The string representation of the current BigInteger value. </returns> public override string ToString() { if (BigInteger.Equals(this, Zero)) { return("0"); } var result = new System.Text.StringBuilder(); var value = this; if (value.Sign < 0) { result.Append('-'); value.sign = 1; } // Overestimate of Floor(Log10(value)) int log10 = (int)Math.Floor(Log(value, 10)) + 1; var divisor = Pow(10, log10); // Adjust the values so that Quorum works. SetupQuorum(ref value, ref divisor); // Check for overestimate of log10. if (BigInteger.Compare(divisor, value) > 0) { value = BigInteger.MultiplyAdd(value, 10, 0); log10--; } for (int i = 0; i <= log10; i++) { // value = value / divisor int digit = Quorem(ref value, divisor); // Append the digit. result.Append((char)(digit + '0')); // value = value * 10; value = BigInteger.MultiplyAdd(value, 10, 0); } return(result.ToString()); }
/// <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> /// Converts a string to an integer (used by parseInt). /// </summary> /// <param name="radix"> The numeric base to use for parsing. Pass zero to use base 10 /// except when the input string starts with '0' in which case base 16 or base 8 are used /// instead. </param> /// <param name="allowOctal"> <c>true</c> if numbers with a leading zero should be parsed /// as octal numbers. </param> /// <returns> The result of parsing the string as a integer. </returns> internal static double ParseInt(string input, int radix, bool allowOctal) { var reader = new System.IO.StringReader(input); int digitCount = 0; // Skip whitespace and line terminators. while (IsWhiteSpaceOrLineTerminator(reader.Peek())) { reader.Read(); } // Determine the sign. double sign = 1; if (reader.Peek() == '+') { reader.Read(); } else if (reader.Peek() == '-') { sign = -1; reader.Read(); } // Hex prefix should be stripped if the radix is 0, undefined or 16. bool stripPrefix = radix == 0 || radix == 16; // Default radix is 10. if (radix == 0) { radix = 10; } // Skip past the prefix, if there is one. if (stripPrefix == true) { if (reader.Peek() == '0') { reader.Read(); digitCount = 1; // Note: required for parsing "0z11" correctly (when radix = 0). int c = reader.Peek(); if (c == 'x' || c == 'X') { // Hex number. reader.Read(); radix = 16; } if (c >= '0' && c <= '9' && allowOctal == true) { // Octal number. radix = 8; } } } // Calculate the maximum number of digits before arbitrary precision arithmetic is // required. int maxDigits = (int)Math.Floor(53 / Math.Log(radix, 2)); // Read numeric digits 0-9, a-z or A-Z. double result = 0; var bigResult = BigInteger.Zero; while (true) { int numericValue = -1; int c = reader.Read(); if (c >= '0' && c <= '9') { numericValue = c - '0'; } if (c >= 'a' && c <= 'z') { numericValue = c - 'a' + 10; } if (c >= 'A' && c <= 'Z') { numericValue = c - 'A' + 10; } if (numericValue == -1 || numericValue >= radix) { break; } if (digitCount == maxDigits) { bigResult = BigInteger.FromDouble(result); } result = result * radix + numericValue; if (digitCount >= maxDigits) { bigResult = BigInteger.MultiplyAdd(bigResult, radix, numericValue); } digitCount++; } // If the input is empty, then return NaN. if (digitCount == 0) { return(double.NaN); } // Numbers with lots of digits require the use of arbitrary precision arithmetic to // determine the correct answer. if (digitCount > maxDigits) { return(RefineEstimate(result, 0, bigResult) * sign); } return(result * sign); }
/// <summary> /// Parses a number and returns the corresponding double-precision value. /// </summary> /// <param name="reader"> The reader to read characters from. </param> /// <param name="firstChar"> The first character of the number. Must be 0-9 or a period. </param> /// <param name="status"> Upon returning, contains the type of error if one occurred. </param> /// <param name="allowHex"> </param> /// <param name="allowOctal"> </param> /// <returns> The numeric value, or <c>NaN</c> if the number is invalid. </returns> internal static double ParseCore(TextReader reader, char firstChar, out ParseCoreStatus status, bool allowHex = true, bool allowOctal = true) { double result; // A count of the number of integral and fractional digits of the input number. int totalDigits = 0; // Assume success. status = ParseCoreStatus.Success; // If the number starts with '0' then the number is a hex literal or a octal literal. if (firstChar == '0' && (allowHex == true || allowOctal == true)) { // Read the next char - should be 'x' or 'X' if this is a hex number (could be just '0'). int c = reader.Peek(); if ((c == 'x' || c == 'X') && allowHex == true) { // Hex number. reader.Read(); result = ParseHex(reader); if (double.IsNaN(result) == true) { status = ParseCoreStatus.InvalidHexLiteral; return(double.NaN); } status = ParseCoreStatus.HexLiteral; return(result); } else if (c >= '0' && c <= '9' && allowOctal == true) { // Octal number. result = ParseOctal(reader); if (double.IsNaN(result) == true) { status = ParseCoreStatus.InvalidOctalLiteral; return(double.NaN); } status = ParseCoreStatus.OctalLiteral; return(result); } } // desired1-3 hold the integral and fractional digits of the input number. // desired1 holds the first set of nine digits, desired2 holds the second set of nine // digits, desired3 holds the rest. int desired1 = 0; int desired2 = 0; var desired3 = BigInteger.Zero; // Indicates the base-10 scale factor of the output e.g. the result is // desired x 10^exponentBase10. int exponentBase10 = 0; // Read the integer component. if (firstChar >= '0' && firstChar <= '9') { desired1 = firstChar - '0'; totalDigits = 1; while (true) { int c = reader.Peek(); if (c < '0' || c > '9') { break; } reader.Read(); if (totalDigits < 9) { desired1 = desired1 * 10 + (c - '0'); } else if (totalDigits < 18) { desired2 = desired2 * 10 + (c - '0'); } else { desired3 = BigInteger.MultiplyAdd(desired3, 10, c - '0'); } totalDigits++; } } if (firstChar == '.' || reader.Peek() == '.') { // Skip past the period. if (firstChar != '.') { reader.Read(); } // Read the fractional component. int fractionalDigits = 0; while (true) { int c = reader.Peek(); if (c < '0' || c > '9') { break; } reader.Read(); if (totalDigits < 9) { desired1 = desired1 * 10 + (c - '0'); } else if (totalDigits < 18) { desired2 = desired2 * 10 + (c - '0'); } else { desired3 = BigInteger.MultiplyAdd(desired3, 10, c - '0'); } totalDigits++; fractionalDigits++; exponentBase10--; } // Check if the number consists solely of a period. if (totalDigits == 0) { status = ParseCoreStatus.NoDigits; return(double.NaN); } // Check if the number has a period but no digits afterwards. if (fractionalDigits == 0) { status = ParseCoreStatus.NoFraction; } } if (reader.Peek() == 'e' || reader.Peek() == 'E') { // Skip past the 'e'. reader.Read(); // Read the sign of the exponent. bool exponentNegative = false; int c = reader.Peek(); if (c == '+') { reader.Read(); } else if (c == '-') { reader.Read(); exponentNegative = true; } // Read the first character of the exponent. int firstExponentChar = reader.Read(); // Check there is a number after the 'e'. int exponent = 0; if (firstExponentChar < '0' || firstExponentChar > '9') { status = ParseCoreStatus.NoExponent; } else { // Read the rest of the exponent. exponent = firstExponentChar - '0'; int exponentDigits = 1; while (true) { c = reader.Peek(); if (c < '0' || c > '9') { break; } reader.Read(); exponent = Math.Min(exponent * 10 + (c - '0'), 9999); exponentDigits++; } // JSON does not allow a leading zero in front of the exponent. if (firstExponentChar == '0' && exponentDigits > 1 && status == ParseCoreStatus.Success) { status = ParseCoreStatus.ExponentHasLeadingZero; } } // Keep track of the overall base-10 exponent. exponentBase10 += exponentNegative ? -exponent : exponent; } // Calculate the integral and fractional portion of the number, scaled to an integer. if (totalDigits < 16) { // Combine desired1 and desired2 to produce an integer representing the final // result. result = (double)((long)desired1 * integerPowersOfTen[Math.Max(totalDigits - 9, 0)] + desired2); } else { // Combine desired1, desired2 and desired3 to produce an integer representing the // final result. var temp = desired3; desired3 = new BigInteger((long)desired1 * integerPowersOfTen[Math.Min(totalDigits - 9, 9)] + desired2); if (totalDigits > 18) { desired3 = BigInteger.Multiply(desired3, BigInteger.Pow(10, totalDigits - 18)); desired3 = BigInteger.Add(desired3, temp); } result = desired3.ToDouble(); } // Apply the base-10 exponent. if (exponentBase10 > 0) { result *= Math.Pow(10, exponentBase10); } else if (exponentBase10 < 0 && exponentBase10 >= -308) { result /= Math.Pow(10, -exponentBase10); } else if (exponentBase10 < -308) { // Note: 10^308 is the largest representable power of ten. result /= Math.Pow(10, 308); result /= Math.Pow(10, -exponentBase10 - 308); } // Numbers with 16 or more digits require the use of arbitrary precision arithmetic to // determine the correct answer. if (totalDigits >= 16) { return(RefineEstimate(result, exponentBase10, desired3)); } return(result); }