public void Register(MathContext context)
{
context.SetNameValue("e", new MathNumber(Math.E));
context.SetNameValue("ln10", new MathNumber(Math.Log(10)));
context.SetNameValue("ln2", new MathNumber(Math.Log(2)));
context.SetNameValue("log2e", new MathNumber(1 / Math.Log(2)));
context.SetNameValue("log10e", new MathNumber(1 / Math.Log(10)));
context.SetNameValue("pi", new MathNumber(Math.PI));
context.SetNameValue("sqrt1_2", new MathNumber(Math.Sqrt(.5)));
context.SetNameValue("sqrt2", new MathNumber(Math.Sqrt(2)));
context.Register("abs", CreateFunction(Math.Abs));
context.Register("acos", CreateFunction(Math.Acos));
context.Register("asin", CreateFunction(Math.Asin));
context.Register("atan", CreateFunction(Math.Atan));
context.Register("ceil", CreateFunction(Math.Ceiling));
context.Register("cos", CreateFunction(Math.Cos));
context.Register("exp", CreateFunction(Math.Exp));
context.Register("floor", CreateFunction(Math.Floor));
context.Register("log", CreateFunction(Math.Log));
context.Register("random", args => new MathNumber(_random.NextDouble()));
context.Register("round", CreateFunction(Math.Round));
context.Register("sin", CreateFunction(Math.Sin));
context.Register("sqrt", CreateFunction(Math.Sqrt));
context.Register("tan", CreateFunction(Math.Tan));
context.Register("atan2", CreateFunction2(Math.Atan2));
context.Register("pow", CreateFunction2(Math.Pow));
}
public PowerTwoNPlusOneIterator(BigDecimal x, MathContext mathContext)
{
this.mathContext = mathContext;
xPowerTwo = x.multiply(x, mathContext);
powerOfX = x;
}
public IScheduler GetCurrentScheduler()
{
MathContext ctx = MathContext.Current;
Guid currentId = ctx.InstanceId;
lock (_schedulerStack)
{
IScheduler scheduler;
if (_schedulerStack.TryGetValue(ctx.InstanceId, out scheduler))
{
return(scheduler);
}
while (ctx.HasParent)
{
ctx = ctx.ParentContext;
if (_schedulerStack.TryGetValue(ctx.InstanceId, out scheduler))
{
return(scheduler);
}
}
scheduler = new ImmediateScheduler();
_schedulerStack.Add(currentId, scheduler);
return(scheduler);
}
}
/// <param name="mContext"> Precision setting for computing the quadrature rules. </param>
public LegendreHighPrecisionRuleFactory(MathContext mContext)
{
this.mContext = mContext;
two = new decimal("2", mContext);
minusOne = new decimal("-1", mContext);
oneHalf = new decimal("0.5", mContext);
}
public PowerTwoNMinusOneIterator(BigDecimal x, MathContext mathContext)
{
this.mathContext = mathContext;
xPowerTwo = x.multiply(x, mathContext);
powerOfX = BigDecimalMath.reciprocal(x, mathContext);
}
public void MathContextStacking()
{
MathContext c1 = MathContext.Current;
Assert.IsNotNull(c1, "01");
Assert.AreNotEqual(Guid.Empty, c1.InstanceId, "02");
Guid id1 = c1.InstanceId;
Assert.AreEqual(id1, MathContext.Current.InstanceId, "03");
Assert.AreEqual(false, c1.HasParent, "04");
Assert.IsNull(c1.ParentContext, "05");
using (MathContext c2 = MathContext.Create())
{
Assert.IsNotNull(c2, "06");
Assert.AreNotEqual(Guid.Empty, c2.InstanceId, "07");
Assert.AreNotEqual(c1.InstanceId, c2.InstanceId, "08");
Assert.AreEqual(true, c2.HasParent, "09");
Assert.AreEqual(c1.InstanceId, c2.ParentContext.InstanceId, "10");
Assert.AreEqual(id1, c1.InstanceId, "11");
Assert.AreEqual(c2.InstanceId, MathContext.Current.InstanceId, "12");
}
Assert.AreEqual(id1, MathContext.Current.InstanceId, "13");
Assert.AreEqual(id1, c1.InstanceId, "14");
}
public BigFloat Divide(BigFloat divisor, MathContext mc)
{
BigFloat ret = this.Divide(divisor);
// TODO: mc?
return(ret);
}
/// <summary>
/// Ctor.
/// </summary>
/// <param name="divisionByZeroReturnsNull">false for division-by-zero returns infinity, true for null</param>
/// <param name="mathContext">math context</param>
public DivideDecimalWMathContext(
bool divisionByZeroReturnsNull,
MathContext mathContext)
{
this._divisionByZeroReturnsNull = divisionByZeroReturnsNull;
this._mathContext = mathContext;
}
/**
* Calculates {@link BigComplex} x to the power of <code>long</code> y (x<sup>y</sup>).
*
* <p>The implementation tries to minimize the number of multiplications of {@link BigComplex x} (using squares whenever possible).</p>
*
* <p>See: <a href="https://en.wikipedia.org/wiki/Exponentiation#Efficient_computation_with_integer_exponents">Wikipedia: Exponentiation - efficient computation</a></p>
*
* @param x the {@link BigComplex} value to take to the power
* @param y the <code>long</code> value to serve as exponent
* @param mathContext the {@link MathContext} used for the result
* @return the calculated x to the power of y with the precision specified in the <code>mathContext</code>
*/
public static BigComplex pow(this BigComplex x, long y, MathContext mathContext)
{
MathContext mc = new MathContext(mathContext.getPrecision() + 10, mathContext.getRoundingMode());
if (y < 0)
{
return(BigComplex.ONE.divide(pow(x, -y, mc), mc).round(mathContext));
}
BigComplex result = BigComplex.ONE;
while (y > 0)
{
if ((y & 1) == 1)
{
// odd exponent -> multiply result with x
result = result.multiply(x, mc);
y -= 1;
}
if (y > 0)
{
// even exponent -> square x
x = x.multiply(x, mc);
}
y >>= 1;
}
return(result.round(mathContext));
}
public PowerNIterator(BigDecimal x, MathContext mathContext)
{
this.x = x;
this.mathContext = mathContext;
powerOfX = BigDecimal.ONE;
}
public PowerTwoNIterator(BigDecimal x, MathContext mathContext)
{
this.mathContext = mathContext;
xPowerTwo = x.multiply(x, mathContext);
powerOfX = BigDecimal.ONE;
}
/// <summary>
/// Ctor
/// </summary>
/// <param name="allowExtendedAggregationFunc">true to allow non-SQL standard builtin agg functions.</param>
/// <param name="isUdfCache">if set to <c>true</c> [is udf cache].</param>
/// <param name="isDuckType">if set to <c>true</c> [is duck type].</param>
/// <param name="sortUsingCollator">if set to <c>true</c> [sort using collator].</param>
/// <param name="optionalDefaultMathContext">The optional default math context.</param>
/// <param name="timeZone"></param>
/// <param name="threadingProfile"></param>
/// <param name="aggregationFactoryFactory"></param>
public EngineImportServiceImpl(
bool allowExtendedAggregationFunc,
bool isUdfCache,
bool isDuckType,
bool sortUsingCollator,
MathContext optionalDefaultMathContext,
TimeZoneInfo timeZone,
ConfigurationEngineDefaults.ThreadingProfile threadingProfile,
AggregationFactoryFactory aggregationFactoryFactory)
{
_imports = new List <AutoImportDesc>();
_annotationImports = new List <AutoImportDesc>();
_aggregationFunctions = new Dictionary <String, ConfigurationPlugInAggregationFunction>();
_aggregationAccess = new List <Pair <ISet <String>, ConfigurationPlugInAggregationMultiFunction> >();
_singleRowFunctions = new Dictionary <String, EngineImportSingleRowDesc>();
_methodInvocationRef = new Dictionary <String, ConfigurationMethodRef>();
_allowExtendedAggregationFunc = allowExtendedAggregationFunc;
IsUdfCache = isUdfCache;
IsDuckType = isDuckType;
_sortUsingCollator = sortUsingCollator;
_optionalDefaultMathContext = optionalDefaultMathContext;
_timeZone = timeZone;
_threadingProfile = threadingProfile;
_aggregationFactoryFactory = aggregationFactoryFactory;
}
public AggregationMethodFactoryAvg(ExprAvgNode parent, Type childType, MathContext optionalMathContext)
{
Parent = parent;
ChildType = childType;
ResultType = GetAvgAggregatorType(childType);
OptionalMathContext = optionalMathContext;
}
private static Computer MakeDecimalComputer(
MathArithTypeEnum operation,
Type typeOne,
Type typeTwo,
bool divisionByZeroReturnsNull,
MathContext optionalMathContext)
{
if (typeOne.IsDecimal() && typeTwo.IsDecimal())
{
if (operation == MathArithTypeEnum.DIVIDE)
{
if (optionalMathContext != null)
{
return(new DivideDecimalWMathContext(divisionByZeroReturnsNull, optionalMathContext));
}
return(new DivideDecimal(divisionByZeroReturnsNull));
}
return(computers.Get(new MathArithDesc(typeof(decimal?), operation)));
}
var convertorOne = SimpleNumberCoercerFactory.GetCoercer(typeOne, typeof(decimal?));
var convertorTwo = SimpleNumberCoercerFactory.GetCoercer(typeTwo, typeof(decimal?));
if (operation == MathArithTypeEnum.ADD)
{
return(new AddDecimalConvComputer(convertorOne, convertorTwo));
}
if (operation == MathArithTypeEnum.SUBTRACT)
{
return(new SubtractDecimalConvComputer(convertorOne, convertorTwo));
}
if (operation == MathArithTypeEnum.MULTIPLY)
{
return(new MultiplyDecimalConvComputer(convertorOne, convertorTwo));
}
if (operation == MathArithTypeEnum.DIVIDE)
{
if (optionalMathContext == null)
{
return(new DivideDecimalConvComputerNoMathCtx(
convertorOne,
convertorTwo,
divisionByZeroReturnsNull));
}
return(new DivideDecimalConvComputerWithMathCtx(
convertorOne,
convertorTwo,
divisionByZeroReturnsNull,
optionalMathContext));
}
return(new ModuloDouble());
}
public AggregationMethodFactory MakeAvg(
StatementExtensionSvcContext statementExtensionSvcContext,
ExprAvgNode exprAvgNode,
Type childType,
MathContext optionalMathContext)
{
return(new AggregationMethodFactoryAvg(exprAvgNode, childType, optionalMathContext));
}
public EnumAverageDecimalScalarForge(
int streamCountIncoming,
MathContext optionalMathContext)
: base(
streamCountIncoming)
{
_optionalMathContext = optionalMathContext;
}
public EnumAverageDecimalEventsForge(
ExprForge innerExpression,
int streamCountIncoming,
MathContext optionalMathContext)
: base(innerExpression, streamCountIncoming)
{
this.optionalMathContext = optionalMathContext;
}
/**
* Calculates the cosine (cosinus) of {@link BigComplex} x in the complex domain.
*
* @param x the {@link BigComplex} to calculate the cosine for
* @param mathContext the {@link MathContext} used for the result
* @return the calculated cosine {@link BigComplex} with the precision specified in the <code>mathContext</code>
*/
public static BigComplex cos(this BigComplex x, MathContext mathContext)
{
MathContext mc = new MathContext(mathContext.getPrecision() + 4, mathContext.getRoundingMode());
return(BigComplex.valueOf(
BigDecimalMath.cos(x.re, mc).multiply(BigDecimalMath.cosh(x.im, mc), mc).round(mathContext),
BigDecimalMath.sin(x.re, mc).multiply(BigDecimalMath.sinh(x.im, mc), mc).negate().round(mathContext)));
}
public AggregationForgeFactoryAvg(ExprAvgNode parent, Type childType, DataInputOutputSerdeForge distinctSerde, MathContext optionalMathContext)
{
this.parent = parent;
this.childType = childType;
this.distinctSerde = distinctSerde;
this.resultType = GetAvgAggregatorType(childType);
this.optionalMathContext = optionalMathContext;
}
public void AddWithContext(string a, int aScale, string b, int bScale, string c, int cScale, int precision, RoundingMode mode)
{
BigDecimal aNumber = new BigDecimal(BigInteger.Parse(a), aScale);
BigDecimal bNumber = new BigDecimal(BigInteger.Parse(b), bScale);
MathContext mc = new MathContext(precision, mode);
BigDecimal result = aNumber.Add(bNumber, mc);
Assert.AreEqual(c, result.ToString(), "incorrect value");
Assert.AreEqual(cScale, result.Scale, "incorrect scale");
}
public DivideDecimalConvComputerWithMathCtx(
Coercer convOne,
Coercer convTwo,
bool divisionByZeroReturnsNull,
MathContext mathContext)
: base(convOne, convTwo, divisionByZeroReturnsNull)
{
_mathContext = mathContext;
}
public override MathValue Evaluate(MathContext context)
{
var function = context.Get(_name) as MathFunction;
if (function == null)
throw new MathException("Invalid function name: " + _name);
MathValue[] values = _args.Select(arg => arg.Evaluate(context)).ToArray();
return function.Function(values);
}
public override MathValue Evaluate(MathContext context)
{
double value = Child.Evaluate(context).ToNumber();
if (double.IsNaN(value))
return new MathNumber(double.NaN);
return new MathNumber(-value);
}
public void CreateSimulationApartement(IScheduler scheduler)
{
MathContext ctx = MathContext.Current;
lock (_schedulerStack) // don't interfer with GetCurrentScheduler
{
_schedulerStack.Add(ctx.InstanceId, scheduler);
}
}
public static BigDecimal FahrenheitToKexle(BigDecimal fahrenheit)
{
MathContext mcInt = new MathContext(9999, RoundingMode.Ceiling);
BigDecimal f2 = fahrenheit.Subtract(27.0, mcInt);
BigDecimal s1f2s3 = f2.Multiply(0.6, mcInt);
BigDecimal s1f2 = new BigDecimal(2634.0).Divide(s1f2s3, mcInt);
return(s1f2.Subtract(4.0, mcInt));
}
/// <summary>
/// Divides decimals.
/// </summary>
/// <param name="d1">The d1.</param>
/// <param name="d2">The d2.</param>
/// <param name="optionalMathContext">The optional math context.</param>
/// <returns></returns>
public static object DivideDecimalUnchecked(object d1, object d2, MathContext optionalMathContext)
{
var nd1 = d1.AsDecimal();
var nd2 = d2.AsDecimal();
var ndx = nd1 / nd2;
return(optionalMathContext != null
? Math.Round(ndx, optionalMathContext.Precision, optionalMathContext.RoundingMode)
: ndx);
}
/**
* Calculates the natural logarithm of {@link BigComplex} x in the complex domain.
*
* <p>See: <a href="https://en.wikipedia.org/wiki/Complex_logarithm">Wikipedia: Complex logarithm</a></p>
*
* @param x the {@link BigComplex} to calculate the natural logarithm for
* @param mathContext the {@link MathContext} used for the result
* @return the calculated natural logarithm {@link BigComplex} with the precision specified in the <code>mathContext</code>
*/
public static BigComplex log(this BigComplex x, MathContext mathContext)
{
// https://en.wikipedia.org/wiki/Complex_logarithm
MathContext mc1 = new MathContext(mathContext.getPrecision() + 20, mathContext.getRoundingMode());
MathContext mc2 = new MathContext(mathContext.getPrecision() + 5, mathContext.getRoundingMode());
return(BigComplex.valueOf(
BigDecimalMath.log(x.abs(mc1), mc1).round(mathContext),
x.angle(mc2)).round(mathContext));
}
/**
* Calculates {@link BigComplex} x to the power of {@link BigDecimal} y (x<sup>y</sup>).
*
* @param x the {@link BigComplex} value to take to the power
* @param y the {@link BigDecimal} value to serve as exponent
* @param mathContext the {@link MathContext} used for the result
* @return the calculated x to the power of y with the precision specified in the <code>mathContext</code>
*/
public static BigComplex pow(this BigComplex x, BigDecimal y, MathContext mathContext)
{
MathContext mc = new MathContext(mathContext.getPrecision() + 4, mathContext.getRoundingMode());
BigDecimal angleTimesN = x.angle(mc).multiply(y, mc);
return(BigComplex.valueOf(
BigDecimalMath.cos(angleTimesN, mc),
BigDecimalMath.sin(angleTimesN, mc)).multiply(BigDecimalMath.pow(x.abs(mc), y, mc), mc).round(mathContext));
}
public EnumAverageDecimalScalarLambdaForge(
ExprForge innerExpression,
int streamCountIncoming,
ObjectArrayEventType resultEventType,
MathContext optionalMathContext)
: base(innerExpression, streamCountIncoming)
{
this.resultEventType = resultEventType;
this.optionalMathContext = optionalMathContext;
}
/**
* Calculates the natural exponent of {@link BigComplex} x (e<sup>x</sup>) in the complex domain.
*
* <p>See: <a href="https://en.wikipedia.org/wiki/Exponential_function#Complex_plane">Wikipedia: Exponent (Complex plane)</a></p>
*
* @param x the {@link BigComplex} to calculate the exponent for
* @param mathContext the {@link MathContext} used for the result
* @return the calculated exponent {@link BigComplex} with the precision specified in the <code>mathContext</code>
*/
public static BigComplex exp(this BigComplex x, MathContext mathContext)
{
MathContext mc = new MathContext(mathContext.getPrecision() + 4, mathContext.getRoundingMode());
BigDecimal expRe = BigDecimalMath.exp(x.re, mc);
return(BigComplex.valueOf(
expRe.multiply(BigDecimalMath.cos(x.im, mc), mc).round(mathContext),
expRe.multiply(BigDecimalMath.sin(x.im, mc), mc)).round(mathContext));
}
public AggregationFactoryMethodAvg(
ExprAvgNode parent,
Type childType,
MathContext optionalMathContext)
{
this.parent = parent;
this.childType = childType;
resultType = GetAvgAggregatorType(childType);
this.optionalMathContext = optionalMathContext;
}
/**
* Calculates the square root of {@link BigComplex} x in the complex domain (√x).
*
* <p>See <a href="https://en.wikipedia.org/wiki/Square_root#Square_root_of_an_imaginary_number">Wikipedia: Square root (Square root of an imaginary number)</a></p>
*
* @param x the {@link BigComplex} to calculate the square root for
* @param mathContext the {@link MathContext} used for the result
* @return the calculated square root {@link BigComplex} with the precision specified in the <code>mathContext</code>
*/
public static BigComplex sqrt(this BigComplex x, MathContext mathContext)
{
// https://math.stackexchange.com/questions/44406/how-do-i-get-the-square-root-of-a-complex-number
MathContext mc = new MathContext(mathContext.getPrecision() + 4, mathContext.getRoundingMode());
BigDecimal magnitude = x.abs(mc);
BigComplex a = x.add(magnitude, mc);
return(a.divide(a.abs(mc), mc).multiply(BigDecimalMath.sqrt(magnitude, mc), mc).round(mathContext));
}
public void ParseExpression_CustomOperators_Exception(string expected)
{
MathContext context = new MathContext();
ITokenizer tokenizer = new Tokenizer(expected);
IParser parser = new Parser(tokenizer, context);
LangException exception = Assert.Throws <LangException>(() => parser.Parse());
Assert.AreEqual(LangException.ErrorCode.UNEXPECTED_TOKEN, exception.ErrorType);
}
public TVector Substitute(MathContext context)
{
var evalTerms = new Dictionary <TBasis, IAlgebraicExpression>();
foreach (var term in this.Terms)
{
evalTerms.Add(term.Key, term.Value.Substitute(context));
}
return(this.NewExplicitVectorValue(evalTerms));
}
public BigComplex Inverse(MathContext mc)
{
BigDecimal hyp = Norm();
/* 1/(x+iy)= (x-iy)/(x^2+y^2 */
return new BigComplex(Real.Divide(hyp, mc), Imaginary.Divide(hyp, mc).Negate());
}
public static BigDecimal Hypot(int n, BigDecimal x)
{
/* compute n^2+x^2 in infinite precision
*/
BigDecimal z = (new BigDecimal(n)).Pow(2).Add(x.Pow(2));
/* Truncate to the precision set by x. Absolute error = in z (square of the result) is |2*x*xerr|,
* where the error is 1/2 of the ulp. Two intermediate protection digits.
* zerr is a signed value, but used only in conjunction with err2prec(), so this feature does not harm.
*/
double zerr = x.ToDouble()*x.Ulp().ToDouble();
var mc = new MathContext(2 + ErrorToPrecision(z.ToDouble(), zerr));
/* Pull square root */
z = Sqrt(z.Round(mc));
/* Final rounding. Absolute error in the square root is x*xerr/z, where zerr holds 2*x*xerr.
*/
mc = new MathContext(ErrorToPrecision(z.ToDouble(), 0.5*zerr/z.ToDouble()));
return z.Round(mc);
}
private static BigDecimal BroadhurstBbp(int n, int p, int[] a, MathContext mc)
{
/* Explore the actual magnitude of the result first with a quick estimate.
*/
double x = 0.0;
for (int k = 1; k < 10; k++)
x += a[(k - 1)%8]/System.Math.Pow(2d, p*(k + 1)/2d)/System.Math.Pow(k, n);
/* Convert the relative precision and estimate of the result into an absolute precision.
*/
double eps = PrecisionToError(x, mc.Precision);
/* Divide this through the number of terms in the sum to account for error accumulation
* The divisor 2^(p(k+1)/2) means that on the average each 8th term in k has shrunk by
* relative to the 8th predecessor by 1/2^(4p). 1/2^(4pc) = 10^(-precision) with c the 8term
* cycles yields c=log_2( 10^precision)/4p = 3.3*precision/4p with k=8c
*/
var kmax = (int) (6.6*mc.Precision/p);
/* Now eps is the absolute error in each term */
eps /= kmax;
BigDecimal res = BigDecimal.Zero;
for (int c = 0;; c++) {
var r = new Rational();
for (int k = 0; k < 8; k++) {
var tmp = new Rational(BigInteger.ValueOf(a[k]), (BigInteger.ValueOf((1 + 8*c + k))).Pow(n));
/* floor( (pk+p)/2)
*/
int pk1h = p*(2 + 8*c + k)/2;
tmp = tmp.Divide(BigInteger.One.ShiftLeft(pk1h));
r = r.Add(tmp);
}
if (System.Math.Abs(r.ToDouble()) < eps)
break;
var mcloc = new MathContext(1 + ErrorToPrecision(r.ToDouble(), eps));
res = res.Add(r.ToBigDecimal(mcloc));
}
return res.Round(mc);
}
public static BigDecimal DivideRound(BigDecimal x, int n)
{
// The estimation of the relative error in the result is |err(x)/x|
var mc = new MathContext(x.Precision);
return x.Divide(new BigDecimal(n), mc);
}
public static BigDecimal Cot(BigDecimal x)
{
if (x.CompareTo(BigDecimal.Zero) == 0) {
throw new ArithmeticException("Cannot take cot of zero " + x);
}
if (x.CompareTo(BigDecimal.Zero) < 0) {
return Cot(x.Negate()).Negate();
}
/* reduce modulo pi
*/
BigDecimal res = ModPi(x);
/* absolute error in the result is err(x)/sin^2(x) to lowest order
*/
double xDbl = res.ToDouble();
double xUlpDbl = x.Ulp().ToDouble()/2d;
double eps = xUlpDbl/2d/System.Math.Pow(System.Math.Sin(xDbl), 2d);
BigDecimal xhighpr = ScalePrecision(res, 2);
BigDecimal xhighprSq = MultiplyRound(xhighpr, xhighpr);
var mc = new MathContext(ErrorToPrecision(xhighpr.ToDouble(), eps));
BigDecimal resul = BigDecimal.One.Divide(xhighpr, mc);
/* x^(2i-1) */
BigDecimal xpowi = xhighpr;
var b = new Bernoulli();
/* 2^(2i) */
var fourn = BigInteger.Parse("4");
/* (2i)! */
BigInteger fac = BigInteger.One;
for (int i = 1;; i++) {
Rational f = b[2*i];
fac = fac.Multiply(BigInteger.ValueOf((2*i))).Multiply(BigInteger.ValueOf((2*i - 1)));
f = f.Multiply(fourn).Divide(fac);
BigDecimal c = MultiplyRound(xpowi, f);
if (i%2 == 0)
resul = resul.Add(c);
else
resul = resul.Subtract(c);
if (System.Math.Abs(c.ToDouble()) < 0.1*eps)
break;
fourn = fourn.ShiftLeft(2);
xpowi = MultiplyRound(xpowi, xhighprSq);
}
mc = new MathContext(ErrorToPrecision(resul.ToDouble(), eps));
return resul.Round(mc);
}
public static BigDecimal ModPi(BigDecimal x)
{
/* write x= pi*k+r with the precision in r defined by the precision of x and not
* compromised by the precision of pi, so the ulp of pi*k should match the ulp of x.
* First get a guess of k to figure out how many digits of pi are needed.
*/
var k = (int) (x.ToDouble()/System.Math.PI);
/* want to have err(pi*k)< err(x)=x.ulp/2, so err(pi) = err(x)/(2k) with two safety digits
*/
double errpi;
if (k != 0)
errpi = 0.5*System.Math.Abs(x.Ulp().ToDouble()/k);
else
errpi = 0.5*System.Math.Abs(x.Ulp().ToDouble());
var mc = new MathContext(2 + ErrorToPrecision(3.1416, errpi));
BigDecimal onepi = PiRound(mc);
BigDecimal pihalf = onepi.Divide(new BigDecimal(2));
/* Delegate the actual operation to the BigDecimal class, which may return
* a negative value of x was negative .
*/
BigDecimal res = x.Remainder(onepi);
if (res.CompareTo(pihalf) > 0)
res = res.Subtract(onepi);
else if (res.CompareTo(pihalf.Negate()) < 0)
res = res.Add(onepi);
/* The actual precision is set by the input value, its absolute value of x.ulp()/2.
*/
mc = new MathContext(ErrorToPrecision(res.ToDouble(), x.Ulp().ToDouble()/2d));
return res.Round(mc);
}
public static BigDecimal Exp(MathContext mc)
{
/* look it up if possible */
if (mc.Precision < E.Precision)
return E.Round(mc);
/* Instantiate a 1.0 with the requested pseudo-accuracy
* and delegate the computation to the public method above.
*/
BigDecimal uni = ScalePrecision(BigDecimal.One, mc.Precision);
return Exp(uni);
}
public static BigDecimal Log(Rational r, MathContext mc)
{
/* the value is undefined if x is negative.
*/
if (r.CompareTo(Rational.Zero) <= 0)
throw new ArithmeticException("Cannot take log of negative " + r);
if (r.CompareTo(Rational.One) == 0)
return BigDecimal.Zero;
/* log(r+epsr) = log(r)+epsr/r. Convert the precision to an absolute error in the result.
* eps contains the required absolute error of the result, epsr/r.
*/
double eps = PrecisionToError(System.Math.Log(r.ToDouble()), mc.Precision);
/* Convert this further into a requirement of the relative precision in r, given that
* epsr/r is also the relative precision of r. Add one safety digit.
*/
var mcloc = new MathContext(1 + ErrorToPrecision(eps));
BigDecimal resul = Log(r.ToBigDecimal(mcloc));
return resul.Round(mc);
}
public static BigDecimal Mod2Pi(BigDecimal x)
{
/* write x= 2*pi*k+r with the precision in r defined by the precision of x and not
* compromised by the precision of 2*pi, so the ulp of 2*pi*k should match the ulp of x.
* First get a guess of k to figure out how many digits of 2*pi are needed.
*/
var k = (int) (0.5*x.ToDouble()/System.Math.PI);
/* want to have err(2*pi*k)< err(x)=0.5*x.ulp, so err(pi) = err(x)/(4k) with two safety digits
*/
double err2pi;
if (k != 0)
err2pi = 0.25*System.Math.Abs(x.Ulp().ToDouble()/k);
else
err2pi = 0.5*System.Math.Abs(x.Ulp().ToDouble());
var mc = new MathContext(2 + ErrorToPrecision(6.283, err2pi));
BigDecimal twopi = PiRound(mc).Multiply(new BigDecimal(2));
/* Delegate the actual operation to the BigDecimal class, which may return
* a negative value of x was negative .
*/
BigDecimal res = x.Remainder(twopi);
if (res.CompareTo(BigDecimal.Zero) < 0)
res = res.Add(twopi);
/* The actual precision is set by the input value, its absolute value of x.ulp()/2.
*/
mc = new MathContext(ErrorToPrecision(res.ToDouble(), x.Ulp().ToDouble()/2d));
return res.Round(mc);
}
public static BigDecimal Log(int n, MathContext mc)
{
/* the value is undefined if x is negative.
*/
if (n <= 0)
throw new ArithmeticException("Cannot take log of negative " + n);
if (n == 1)
return BigDecimal.Zero;
if (n == 2) {
if (mc.Precision < Log2.Precision)
return Log2.Round(mc);
/* Broadhurst <a href="http://arxiv.org/abs/math/9803067">arXiv:math/9803067</a>
* Error propagation: the error in log(2) is twice the error in S(2,-5,...).
*/
int[] a = {2, -5, -2, -7, -2, -5, 2, -3};
BigDecimal S = BroadhurstBbp(2, 1, a, new MathContext(1 + mc.Precision));
S = S.Multiply(new BigDecimal(8));
S = Sqrt(DivideRound(S, 3));
return S.Round(mc);
}
if (n == 3) {
/* summation of a series roughly proportional to (7/500)^k. Estimate count
* of terms to estimate the precision (drop the favorable additional
* 1/k here): 0.013^k <= 10^(-precision), so k*log10(0.013) <= -precision
* so k>= precision/1.87.
*/
var kmax = (int) (mc.Precision/1.87);
var mcloc = new MathContext(mc.Precision + 1 + (int) (System.Math.Log10(kmax*0.693/1.098)));
BigDecimal log3 = MultiplyRound(Log(2, mcloc), 19);
/* log3 is roughly 1, so absolute and relative error are the same. The
* result will be divided by 12, so a conservative error is the one
* already found in mc
*/
double eps = PrecisionToError(1.098, mc.Precision)/kmax;
var r = new Rational(7153, 524288);
var pk = new Rational(7153, 524288);
for (int k = 1;; k++) {
Rational tmp = pk.Divide(k);
if (tmp.ToDouble() < eps)
break;
/* how many digits of tmp do we need in the sum?
*/
mcloc = new MathContext(ErrorToPrecision(tmp.ToDouble(), eps));
BigDecimal c = pk.Divide(k).ToBigDecimal(mcloc);
if (k%2 != 0)
log3 = log3.Add(c);
else
log3 = log3.Subtract(c);
pk = pk.Multiply(r);
}
log3 = DivideRound(log3, 12);
return log3.Round(mc);
}
if (n == 5) {
/* summation of a series roughly proportional to (7/160)^k. Estimate count
* of terms to estimate the precision (drop the favorable additional
* 1/k here): 0.046^k <= 10^(-precision), so k*log10(0.046) <= -precision
* so k>= precision/1.33.
*/
var kmax = (int) (mc.Precision/1.33);
var mcloc = new MathContext(mc.Precision + 1 + (int) (System.Math.Log10(kmax*0.693/1.609)));
BigDecimal log5 = MultiplyRound(Log(2, mcloc), 14);
/* log5 is roughly 1.6, so absolute and relative error are the same. The
* result will be divided by 6, so a conservative error is the one
* already found in mc
*/
double eps = PrecisionToError(1.6, mc.Precision)/kmax;
var r = new Rational(759, 16384);
var pk = new Rational(759, 16384);
for (int k = 1;; k++) {
Rational tmp = pk.Divide(k);
if (tmp.ToDouble() < eps)
break;
/* how many digits of tmp do we need in the sum?
*/
mcloc = new MathContext(ErrorToPrecision(tmp.ToDouble(), eps));
BigDecimal c = pk.Divide(k).ToBigDecimal(mcloc);
log5 = log5.Subtract(c);
pk = pk.Multiply(r);
}
log5 = DivideRound(log5, 6);
return log5.Round(mc);
}
if (n == 7) {
/* summation of a series roughly proportional to (1/8)^k. Estimate count
* of terms to estimate the precision (drop the favorable additional
* 1/k here): 0.125^k <= 10^(-precision), so k*log10(0.125) <= -precision
* so k>= precision/0.903.
*/
var kmax = (int) (mc.Precision/0.903);
var mcloc = new MathContext(mc.Precision + 1 + (int) (System.Math.Log10(kmax*3*0.693/1.098)));
BigDecimal log7 = MultiplyRound(Log(2, mcloc), 3);
/* log7 is roughly 1.9, so absolute and relative error are the same.
*/
double eps = PrecisionToError(1.9, mc.Precision)/kmax;
var r = new Rational(1, 8);
var pk = new Rational(1, 8);
for (int k = 1;; k++) {
Rational tmp = pk.Divide(k);
if (tmp.ToDouble() < eps)
break;
/* how many digits of tmp do we need in the sum?
*/
mcloc = new MathContext(ErrorToPrecision(tmp.ToDouble(), eps));
BigDecimal c = pk.Divide(k).ToBigDecimal(mcloc);
log7 = log7.Subtract(c);
pk = pk.Multiply(r);
}
return log7.Round(mc);
} else {
/* At this point one could either forward to the log(BigDecimal) signature (implemented)
* or decompose n into Ifactors and use an implemenation of all the prime bases.
* Estimate of the result; convert the mc argument to an absolute error eps
* log(n+errn) = log(n)+errn/n = log(n)+eps
*/
double res = System.Math.Log(n);
double eps = PrecisionToError(res, mc.Precision);
/* errn = eps*n, convert absolute error in result to requirement on absolute error in input
*/
eps *= n;
/* Convert this absolute requirement of error in n to a relative error in n
*/
var mcloc = new MathContext(1 + ErrorToPrecision(n, eps));
/* Padd n with a number of zeros to trigger the required accuracy in
* the standard signature method
*/
BigDecimal nb = ScalePrecision(new BigDecimal(n), mcloc);
return Log(nb);
}
}
public static BigDecimal Log(BigDecimal x)
{
/* the value is undefined if x is negative.
*/
if (x.CompareTo(BigDecimal.Zero) < 0)
throw new ArithmeticException("Cannot take log of negative " + x);
if (x.CompareTo(BigDecimal.One) == 0) {
/* log 1. = 0. */
return ScalePrecision(BigDecimal.Zero, x.Precision - 1);
}
if (System.Math.Abs(x.ToDouble() - 1.0) <= 0.3) {
/* The standard Taylor series around x=1, z=0, z=x-1. Abramowitz-Stegun 4.124.
* The absolute error is err(z)/(1+z) = err(x)/x.
*/
BigDecimal z = ScalePrecision(x.Subtract(BigDecimal.One), 2);
BigDecimal zpown = z;
double eps = 0.5*x.Ulp().ToDouble()/System.Math.Abs(x.ToDouble());
BigDecimal resul = z;
for (int k = 2;; k++) {
zpown = MultiplyRound(zpown, z);
BigDecimal c = DivideRound(zpown, k);
if (k%2 == 0)
resul = resul.Subtract(c);
else
resul = resul.Add(c);
if (System.Math.Abs(c.ToDouble()) < eps)
break;
}
var mc = new MathContext(ErrorToPrecision(resul.ToDouble(), eps));
return resul.Round(mc);
} else {
double xDbl = x.ToDouble();
double xUlpDbl = x.Ulp().ToDouble();
/* Map log(x) = log root[r](x)^r = r*log( root[r](x)) with the aim
* to move roor[r](x) near to 1.2 (that is, below the 0.3 appearing above), where log(1.2) is roughly 0.2.
*/
var r = (int) (System.Math.Log(xDbl)/0.2);
/* Since the actual requirement is a function of the value 0.3 appearing above,
* we avoid the hypothetical case of endless recurrence by ensuring that r >= 2.
*/
r = System.Math.Max(2, r);
/* Compute r-th root with 2 additional digits of precision
*/
BigDecimal xhighpr = ScalePrecision(x, 2);
BigDecimal resul = Root(r, xhighpr);
resul = Log(resul).Multiply(new BigDecimal(r));
/* error propagation: log(x+errx) = log(x)+errx/x, so the absolute error
* in the result equals the relative error in the input, xUlpDbl/xDbl .
*/
var mc = new MathContext(ErrorToPrecision(resul.ToDouble(), xUlpDbl/xDbl));
return resul.Round(mc);
}
}
public static BigComplex InvertRound(BigComplex z)
{
if (z.Imaginary.CompareTo(BigDecimal.Zero) == 0) {
/* In this case with vanishing Im(x), the result is simply 1/Re z.
*/
MathContext mc = new MathContext(z.Real.Precision);
return new BigComplex(BigDecimal.One.Divide(z.Real, mc));
} else if (z.Real.CompareTo(BigDecimal.Zero) == 0) {
/* In this case with vanishing Re(z), the result is simply -i/Im z
*/
MathContext mc = new MathContext(z.Imaginary.Precision);
return new BigComplex(BigDecimal.Zero, BigDecimal.One.Divide(z.Imaginary, mc).Negate());
} else {
/* 1/(x.re+I*x.im) = 1/(x.re+x.im^2/x.re) - I /(x.im +x.re^2/x.im)
*/
BigDecimal R = AddRound(z.Real, DivideRound(MultiplyRound(z.Imaginary, z.Imaginary), z.Real));
BigDecimal I = AddRound(z.Imaginary, DivideRound(MultiplyRound(z.Real, z.Real), z.Imaginary));
MathContext mc = new MathContext(1 + R.Precision);
R = BigDecimal.One.Divide(R, mc);
mc = new MathContext(1 + I.Precision);
I = BigDecimal.One.Divide(I, mc);
return new BigComplex(R, I.Negate());
}
}
public BigComplex Multiply(BigComplex oth, MathContext mc)
{
BigDecimal a = Real.Add(Imaginary).Multiply(oth.Real);
BigDecimal b = oth.Real.Add(oth.Imaginary).Multiply(Imaginary);
BigDecimal c = oth.Imaginary.Subtract(oth.Real).Multiply(Real);
BigDecimal x = a.Subtract(b, mc);
BigDecimal y = a.Add(c, mc);
return new BigComplex(x, y);
}
public static BigDecimal MultiplyRound(BigDecimal x, BigInteger n)
{
BigDecimal resul = x.Multiply(new BigDecimal(n));
// The estimation of the absolute error in the result is |n*err(x)|
var mc = new MathContext(n.CompareTo(BigInteger.Zero) != 0 ? x.Precision : 0);
return resul.Round(mc);
}
public static BigDecimal GammaRound(MathContext mc)
{
if (mc.Precision < Gamma.Precision)
return Gamma.Round(mc);
double eps = PrecisionToError(0.577, mc.Precision);
// Euler-Stieltjes as shown in Dilcher, Aequat Math 48 (1) (1994) 55-85
var mcloc = new MathContext(2 + mc.Precision);
BigDecimal resul = BigDecimal.One;
resul = resul.Add(Log(2, mcloc));
resul = resul.Subtract(Log(3, mcloc));
// how many terms: zeta-1 falls as 1/2^(2n+1), so the
// terms drop faster than 1/2^(4n+2). Set 1/2^(4kmax+2) < eps.
// Leading term zeta(3)/(4^1*3) is 0.017. Leading zeta(3) is 1.2. Log(2) is 0.7
var kmax = (int) ((System.Math.Log(eps/0.7) - 2.0)/4d);
mcloc = new MathContext(1 + ErrorToPrecision(1.2, eps/kmax));
for (int n = 1;; n++) {
// zeta is close to 1. Division of zeta-1 through
// 4^n*(2n+1) means divion through roughly 2^(2n+1)
BigDecimal c = Zeta(2*n + 1, mcloc).Subtract(BigDecimal.One);
BigInteger fourn = BigInteger.ValueOf(2*n + 1);
fourn = fourn.ShiftLeft(2*n);
c = DivideRound(c, fourn);
resul = resul.Subtract(c);
if (c.ToDouble() < 0.1*eps)
break;
}
return resul.Round(mc);
}
public static BigDecimal MultiplyRound(BigDecimal x, BigDecimal y)
{
BigDecimal resul = x.Multiply(y);
// The estimation of the relative error in the result is the sum of the relative
// errors |err(y)/y|+|err(x)/x|
var mc = new MathContext(System.Math.Min(x.Precision, y.Precision));
return resul.Round(mc);
}
public override MathValue Evaluate(MathContext context)
{
return context.Get(_name);
}
public static BigDecimal MultiplyRound(BigDecimal x, Rational f)
{
if (f.CompareTo(BigInteger.Zero) == 0)
return BigDecimal.Zero;
/* Convert the rational value with two digits of extra precision
*/
var mc = new MathContext(2 + x.Precision);
BigDecimal fbd = f.ToBigDecimal(mc);
/* and the precision of the product is then dominated by the precision in x
*/
return MultiplyRound(x, fbd);
}
public static BigDecimal DivideRound(BigDecimal x, BigDecimal y)
{
/* The estimation of the relative error in the result is |err(y)/y|+|err(x)/x|
*/
var mc = new MathContext(System.Math.Min(x.Precision, y.Precision));
BigDecimal resul = x.Divide(y, mc);
/* If x and y are precise integer values that may have common factors,
* the method above will truncate trailing zeros, which may result in
* a smaller apparent accuracy than starte... add missing trailing zeros now.
*/
return ScalePrecision(resul, mc);
}
public BigComplex Sqrt(MathContext mc)
{
BigDecimal half = new BigDecimal(2);
/* compute l=sqrt(re^2+im^2), then u=sqrt((l+re)/2)
* and v= +- sqrt((l-re)/2 as the new real and imaginary parts.
*/
BigDecimal l = Abs(mc);
if (l.CompareTo(BigDecimal.Zero) == 0)
return new BigComplex(BigMath.ScalePrecision(BigDecimal.Zero, mc),
BigMath.ScalePrecision(BigDecimal.Zero, mc));
BigDecimal u = BigMath.Sqrt(l.Add(Real).Divide(half, mc), mc);
BigDecimal v = BigMath.Sqrt(l.Subtract(Real).Divide(half, mc), mc);
if (Imaginary.CompareTo(BigDecimal.Zero) >= 0)
return new BigComplex(u, v);
else
return new BigComplex(u, v.Negate());
}
public static BigDecimal Cos(BigDecimal x)
{
if (x.CompareTo(BigDecimal.Zero) < 0)
return Cos(x.Negate());
if (x.CompareTo(BigDecimal.Zero) == 0)
return BigDecimal.One;
/* reduce modulo 2pi
*/
BigDecimal res = Mod2Pi(x);
double errpi = 0.5*System.Math.Abs(x.Ulp().ToDouble());
var mc = new MathContext(2 + ErrorToPrecision(3.14159, errpi));
BigDecimal p = PiRound(mc);
mc = new MathContext(x.Precision);
if (res.CompareTo(p) > 0) {
/* pi<x<=2pi: cos(x)= - cos(x-pi)
*/
return Cos(SubtractRound(res, p)).Negate();
}
if (res.Multiply(BigDecimal.ValueOf(2)).CompareTo(p) > 0) {
/* pi/2<x<=pi: cos(x)= -cos(pi-x)
*/
return Cos(SubtractRound(p, res)).Negate();
}
/* for the range 0<=x<Pi/2 one could use cos(2x)= 1-2*sin^2(x)
* to split this further, or use the cos up to pi/4 and the sine higher up.
throw new ProviderException("Not implemented: cosine ") ;
*/
if (res.Multiply(BigDecimal.ValueOf(4)).CompareTo(p) > 0) {
/* x>pi/4: cos(x) = sin(pi/2-x)
*/
return Sin(SubtractRound(p.Divide(BigDecimal.ValueOf(2)), res));
}
/* Simple Taylor expansion, sum_{i=0..infinity} (-1)^(..)res^(2i)/(2i)! */
BigDecimal resul = BigDecimal.One;
/* x^i */
BigDecimal xpowi = BigDecimal.One;
/* 2i factorial */
BigInteger ifac = BigInteger.One;
/* The absolute error in the result is the error in x^2/2 which is x times the error in x.
*/
double xUlpDbl = 0.5*res.Ulp().ToDouble()*res.ToDouble();
/* The error in the result is set by the error in x^2/2 itself, xUlpDbl.
* We need at most k terms to push x^(2k+1)/(2k+1)! below this value.
* x^(2k) < xUlpDbl; (2k)*log(x) < log(xUlpDbl);
*/
int k = (int) (System.Math.Log(xUlpDbl)/System.Math.Log(res.ToDouble()))/2;
var mcTay = new MathContext(ErrorToPrecision(1d, xUlpDbl/k));
for (int i = 1;; i++) {
/* TBD: at which precision will 2*i-1 or 2*i overflow?
*/
ifac = ifac.Multiply(BigInteger.ValueOf((2*i - 1)));
ifac = ifac.Multiply(BigInteger.ValueOf((2*i)));
xpowi = xpowi.Multiply(res).Multiply(res).Negate();
BigDecimal corr = xpowi.Divide(new BigDecimal(ifac), mcTay);
resul = resul.Add(corr);
if (corr.Abs().ToDouble() < 0.5*xUlpDbl)
break;
}
/* The error in the result is governed by the error in x itself.
*/
mc = new MathContext(ErrorToPrecision(resul.ToDouble(), xUlpDbl));
return resul.Round(mc);
}
public BigDecimal Abs(MathContext mc)
{
return BigMath.Sqrt(Norm(), mc);
}
public static BigDecimal Asin(BigDecimal x)
{
if (x.CompareTo(BigDecimal.One) > 0 ||
x.CompareTo(BigDecimal.One.Negate()) < 0) {
throw new ArithmeticException("Out of range argument " + x + " of asin");
}
if (x.CompareTo(BigDecimal.Zero) == 0)
return BigDecimal.Zero;
if (x.CompareTo(BigDecimal.One) == 0) {
/* arcsin(1) = pi/2
*/
double errpi = System.Math.Sqrt(x.Ulp().ToDouble());
var mc = new MathContext(ErrorToPrecision(3.14159, errpi));
return PiRound(mc).Divide(new BigDecimal(2));
}
if (x.CompareTo(BigDecimal.Zero) < 0) {
return Asin(x.Negate()).Negate();
}
if (x.ToDouble() > 0.7) {
BigDecimal xCompl = BigDecimal.One.Subtract(x);
double xDbl = x.ToDouble();
double xUlpDbl = x.Ulp().ToDouble()/2d;
double eps = xUlpDbl/2d/System.Math.Sqrt(1d - System.Math.Pow(xDbl, 2d));
BigDecimal xhighpr = ScalePrecision(xCompl, 3);
BigDecimal xhighprV = DivideRound(xhighpr, 4);
BigDecimal resul = BigDecimal.One;
/* x^(2i+1) */
BigDecimal xpowi = BigDecimal.One;
/* i factorial */
BigInteger ifacN = BigInteger.One;
BigInteger ifacD = BigInteger.One;
for (int i = 1;; i++) {
ifacN = ifacN.Multiply(BigInteger.ValueOf((2*i - 1)));
ifacD = ifacD.Multiply(BigInteger.ValueOf(i));
if (i == 1)
xpowi = xhighprV;
else
xpowi = MultiplyRound(xpowi, xhighprV);
BigDecimal c = DivideRound(MultiplyRound(xpowi, ifacN),
ifacD.Multiply(BigInteger.ValueOf((2*i + 1))));
resul = resul.Add(c);
/* series started 1+x/12+... which yields an estimate of the sum's error
*/
if (System.Math.Abs(c.ToDouble()) < xUlpDbl/120d)
break;
}
/* sqrt(2*z)*(1+...)
*/
xpowi = Sqrt(xhighpr.Multiply(new BigDecimal(2)));
resul = MultiplyRound(xpowi, resul);
var mc = new MathContext(resul.Precision);
BigDecimal pihalf = PiRound(mc).Divide(new BigDecimal(2));
mc = new MathContext(ErrorToPrecision(resul.ToDouble(), eps));
return pihalf.Subtract(resul, mc);
} else {
/* absolute error in the result is err(x)/sqrt(1-x^2) to lowest order
*/
double xDbl = x.ToDouble();
double xUlpDbl = x.Ulp().ToDouble()/2d;
double eps = xUlpDbl/2d/System.Math.Sqrt(1d - System.Math.Pow(xDbl, 2d));
BigDecimal xhighpr = ScalePrecision(x, 2);
BigDecimal xhighprSq = MultiplyRound(xhighpr, xhighpr);
BigDecimal resul = xhighpr.Plus();
/* x^(2i+1) */
BigDecimal xpowi = xhighpr;
/* i factorial */
BigInteger ifacN = BigInteger.One;
BigInteger ifacD = BigInteger.One;
for (int i = 1;; i++) {
ifacN = ifacN.Multiply(BigInteger.ValueOf((2*i - 1)));
ifacD = ifacD.Multiply(BigInteger.ValueOf((2*i)));
xpowi = MultiplyRound(xpowi, xhighprSq);
BigDecimal c = DivideRound(MultiplyRound(xpowi, ifacN),
ifacD.Multiply(BigInteger.ValueOf((2*i + 1))));
resul = resul.Add(c);
if (System.Math.Abs(c.ToDouble()) < 0.1*eps)
break;
}
var mc = new MathContext(ErrorToPrecision(resul.ToDouble(), eps));
return resul.Round(mc);
}
}
public BigComplex Divide(BigComplex oth, MathContext mc)
{
/* lazy implementation: (x+iy)/(a+ib)= (x+iy)* 1/(a+ib) */
return Multiply(oth.Inverse(mc), mc);
}
public static BigDecimal Exp(BigDecimal x)
{
/* To calculate the value if x is negative, use exp(-x) = 1/exp(x)
*/
if (x.CompareTo(BigDecimal.Zero) < 0) {
BigDecimal invx = Exp(x.Negate());
/* Relative error in inverse of invx is the same as the relative errror in invx.
* This is used to define the precision of the result.
*/
var mc = new MathContext(invx.Precision);
return BigDecimal.One.Divide(invx, mc);
}
if (x.CompareTo(BigDecimal.Zero) == 0) {
/* recover the valid number of digits from x.ulp(), if x hits the
* zero. The x.precision() is 1 then, and does not provide this information.
*/
return ScalePrecision(BigDecimal.One, -(int) (System.Math.Log10(x.Ulp().ToDouble())));
}
/* Push the number in the Taylor expansion down to a small
* value where TAYLOR_NTERM terms will do. If x<1, the n-th term is of the order
* x^n/n!, and equal to both the absolute and relative error of the result
* since the result is close to 1. The x.ulp() sets the relative and absolute error
* of the result, as estimated from the first Taylor term.
* We want x^TAYLOR_NTERM/TAYLOR_NTERM! < x.ulp, which is guaranteed if
* x^TAYLOR_NTERM < TAYLOR_NTERM*(TAYLOR_NTERM-1)*...*x.ulp.
*/
double xDbl = x.ToDouble();
double xUlpDbl = x.Ulp().ToDouble();
if (System.Math.Pow(xDbl, TaylorNterm) < TaylorNterm*(TaylorNterm - 1.0)*(TaylorNterm - 2.0)*xUlpDbl) {
/* Add TAYLOR_NTERM terms of the Taylor expansion (Euler's sum formula)
*/
BigDecimal resul = BigDecimal.One;
/* x^i */
BigDecimal xpowi = BigDecimal.One;
/* i factorial */
BigInteger ifac = BigInteger.One;
/* TAYLOR_NTERM terms to be added means we move x.ulp() to the right
* for each power of 10 in TAYLOR_NTERM, so the addition won't add noise beyond
* what's already in x.
*/
var mcTay = new MathContext(ErrorToPrecision(1d, xUlpDbl/TaylorNterm));
for (int i = 1; i <= TaylorNterm; i++) {
ifac = ifac.Multiply(BigInteger.ValueOf(i));
xpowi = xpowi.Multiply(x);
BigDecimal c = xpowi.Divide(new BigDecimal(ifac), mcTay);
resul = resul.Add(c);
if (System.Math.Abs(xpowi.ToDouble()) < i &&
System.Math.Abs(c.ToDouble()) < 0.5*xUlpDbl)
break;
}
/* exp(x+deltax) = exp(x)(1+deltax) if deltax is <<1. So the relative error
* in the result equals the absolute error in the argument.
*/
var mc = new MathContext(ErrorToPrecision(xUlpDbl/2d));
return resul.Round(mc);
} else {
/* Compute exp(x) = (exp(0.1*x))^10. Division by 10 does not lead
* to loss of accuracy.
*/
var exSc = (int) (1.0 - System.Math.Log10(TaylorNterm*(TaylorNterm - 1.0)*(TaylorNterm - 2.0)*xUlpDbl
/System.Math.Pow(xDbl, TaylorNterm))/(TaylorNterm - 1.0));
BigDecimal xby10 = x.ScaleByPowerOfTen(-exSc);
BigDecimal expxby10 = Exp(xby10);
/* Final powering by 10 means that the relative error of the result
* is 10 times the relative error of the base (First order binomial expansion).
* This looses one digit.
*/
var mc = new MathContext(expxby10.Precision - exSc);
/* Rescaling the powers of 10 is done in chunks of a maximum of 8 to avoid an invalid operation
* response by the BigDecimal.pow library or integer overflow.
*/
while (exSc > 0) {
int exsub = System.Math.Min(8, exSc);
exSc -= exsub;
var mctmp = new MathContext(expxby10.Precision - exsub + 2);
int pex = 1;
while (exsub-- > 0)
pex *= 10;
expxby10 = expxby10.Pow(pex, mctmp);
}
return expxby10.Round(mc);
}
}
public static BigDecimal Hypot(BigDecimal x, BigDecimal y)
{
/* compute x^2+y^2
*/
BigDecimal z = x.Pow(2).Add(y.Pow(2));
/* truncate to the precision set by x and y. Absolute error = 2*x*xerr+2*y*yerr,
* where the two errors are 1/2 of the ulp's. Two intermediate protectio digits.
*/
BigDecimal zerr = x.Abs().Multiply(x.Ulp()).Add(y.Abs().Multiply(y.Ulp()));
var mc = new MathContext(2 + ErrorToPrecision(z, zerr));
/* Pull square root */
z = Sqrt(z.Round(mc));
/* Final rounding. Absolute error in the square root is (y*yerr+x*xerr)/z, where zerr holds 2*(x*xerr+y*yerr).
*/
mc = new MathContext(ErrorToPrecision(z.ToDouble(), 0.5*zerr.ToDouble()/z.ToDouble()));
return z.Round(mc);
}